A Primer on Hilbert Space Operators

The book concisely presents the fundamental aspects of the theory of operators on Hilbert spaces. The topics covered include functional calculus and spectral theorems, compact operators, trace class and Hilbert-Schmidt operators, self-adjoint extensions of symmetric operators, and one-parameter groups of operators. The exposition of the material on unbounded operators is based on a novel tool, called the z-transform, which provides a way to encode full information about unbounded operators in bounded ones, hence making many technical aspects of the theory less involved.


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Compact Textbooks in Mathematics

Piotr Sołtan

A Primer on Hilbert Space Operators

Compact Textbooks in Mathematics

Compact Textbooks in Mathematics This textbook series presents concise introductions to current topics in mathematics and mainly addresses advanced undergraduates and master students. The concept is to offer small books covering subject matter equivalent to 2- or 3-hour lectures or seminars which are also suitable for self-study. The books provide students and teachers with new perspectives and novel approaches. They may feature examples and exercises to illustrate key concepts and applications of the theoretical contents. The series also includes textbooks specifically speaking to the needs of students from other disciplines such as physics, computer science, engineering, life sciences, finance. • compact: small books presenting the relevant knowledge • learning made easy: examples and exercises illustrate the application of the contents • useful for lecturers: each title can serve as basis and guideline for a semester course/lecture/seminar of 2–3 hours per week. More information about this series at http://www.springer.com/series/11225

Piotr Sołtan

A Primer on Hilbert Space Operators

Piotr Sołtan Faculty of Physics University of Warsaw Warsaw, Poland

ISSN 2296-4568 ISSN 2296-455X (electronic) Compact Textbooks in Mathematics ISBN 978-3-319-92060-3 ISBN 978-3-319-92061-0 (eBook) https://doi.org/10.1007/978-3-319-92061-0 Library of Congress Control Number: 2018944130 Mathematics Subject Classification (2010): 47A05, 47A10, 47A60, 47B15, 47B25, 47B10, 47D03 © Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

For Asia, Marcin, and Adam

vii

Preface Theory of operators on Hilbert spaces is one of the great achievements of functional analysis with applications in countless branches of mathematics and physics. It is also the basis of fascinating generalizations which include theory of C∗ -algebras, von Neumann algebras, non-commutative geometry, and many other domains of current research in mathematics. This book is based on extended lecture notes from a course held by the author at the Faculty of Physics of the University of Warsaw. Its main goals are: ▬ To provide a concise presentation of the rudiments of the theory of operators on Hilbert spaces with complete and direct proofs ▬ To prepare the reader for further study of operator theory itself as well as the theory of operator algebras (C∗ -algebras and von Neumann algebras) The extent to which the first goal has been achieved can only be judged having defined the rudiments of the theory of operators on Hilbert spaces which tends to be a highly subjective matter. Some fragments of the theory have been purposefully left out in the hope of keeping the book’s size moderate. More importantly, the reason for restricting the scope of the book to a bare minimum is the existence of a considerable number of unrivaled monographs and textbooks on the subject which cover a vastly larger range of material (e.g., [AkGl, Kat, ReSi1 , ReSi2 ] and particularly [Mau]). However, studying these books might turn out to be quite challenging, and we hope that reading this modest volume might serve as a worthwhile preparatory exercise. The material splits into two main parts. The first one is devoted to studying bounded operators, while the second deals with unbounded ones. In the latter part, we employ a novel and very useful tool introduced into world’s mathematics by S.L. Woronowicz, namely, the so-called z-transform of a closed densely defined operator. Skipping ahead of the detailed introduction of the z-transform in  Chap. 9, we can somewhat imprecisely describe it as a way to encode full information about a closed densely defined operator on a Hilbert space in a bounded operator on this space. The z-transform allows for simple and elegant proofs of many fundamental results of the theory. Most of the excellent textbooks and monographs listed at the end of the book develop spectral theory of operators on Hilbert spaces based on several aspects of the theory of Banach algebras and C∗ -algebras. This approach requires the reader to work with rather sophisticated structures first and only then apply them to more elementary problems of operator theory. Our route is different: spectral theory of operators on Hilbert spaces is presented with minimal use of deeper results of Banach algebras. It is worth mentioning that, in fact, the theory of Banach algebras (in particular C∗ -algebras and von Neumann algebras) grew out of operator theory and can be viewed as its

viii

Preface

generalization par excellence. It is for this reason that we have chosen not to put too much emphasis on various aspects of the theory of Banach algebras.1 It is the hope of the author that this approach will facilitate at least partial achievement of the second of the goals mentioned above. Since the book aims to be a primer on theory of operators on Hilbert spaces, we decided not to include exercises nor many examples. Instead, we placed short notes at the end of each chapter containing references to textbooks and monographs containing a wealth of examples and exercises and, in some instances, information about possible further developments and generalizations of the subject of each chapter. The lecture course on which the book is based was intended for students who have had previous experience with basic functional analysis including rudiments of Banach and Hilbert spaces. In particular, we assume the reader is familiar with: ▬ Basic linear algebra and calculus ▬ Elements of general topology including the concept of a locally compact topological space, a net and its convergence, and the Stone–Weierstrass theorem ▬ Elementary complex analysis including the notion of a holomorphic function, the Cauchy formula, and Liouville’s theorem ▬ Theory of measure and integral including the dominated convergence theorem, product measures and Fubini’s theorem, complex measures, Radon–Nikodym theorem, and Riesz–Markov–Kakutani representation theorem ▬ The concept of a Banach space and Hilbert space, Lp spaces, bounded operators on Banach spaces, and the operator norm ▬ The Riesz representation theorem (for functionals on Hilbert spaces), bounded sesquilinear forms and their relation to bounded operators on Hilbert spaces, and the notion of the adjoint operator of a bounded operator on a Hilbert space. We will also make use of integrals of continuous Banach space-valued functions over compact intervals which are often discussed in courses of ordinary differential equations. A much more general theory of such integrals is presented e.g. in the monograph [Rud2 ]. All of the above topics are covered in standard courses of complex analysis and measure theory, and textbooks such as e.g. [ReSi1 , Rud1 ] discuss most of them. For the reader’s convenience, we have gathered in the Appendix of the book some of the most important tools (inducing classical results such as the Banach-Steinhaus theorem and the closed graph theorem) with complete proofs.

exception from this rule comes in  Chap. 7 in which we apply several results of C∗ -algebra theory. The results in question have been gathered in Appendix A.5.2. 1 One

ix Preface

All vector spaces considered in this book will be over the field of complex numbers. We will employ the conventions of the physics literature according to which scalar products will be linear in the second variable and anti-linear in the first. We will also use the “ket” and “bra” notation which we will now briefly discuss. Let H be a Hilbert space. Then any vector ψ ∈ H defines a unique linear operator C → H mapping 1 ∈ C to ψ. We denote this linear map by the symbol ψ. Now consider on C the standard Hilbert space structure (one for which {1} is an orthonormal basis). Then the adjoint ψ∗ of ψ is a bounded linear functional on H mapping any vector φ onto the number ψ φ. This map is denoted by ψ . In particular, the composition ψ1 ◦ ψ2  is a linear map C → C given by multiplication by the scalar ψ1 ψ2 , while the composition ψ2 ◦ψ1 (customarily written as ψ2  ψ1 ) is a map H → H taking ϕ ∈ H to ψ1 ϕ ψ2 . One of the fundamental formulas from the theory of vector spaces endowed with a sesquilinear form is the polarization formula. It has various, sometimes highly sophisticated, formulations one of which will be especially useful to us: let F be a sesquilinear form on a vector space H (linear with respect to the second argument). Then F (ξ, η) =

1 4

3 

ik F (η + ik ξ, η + ik ξ ),

ξ, η ∈ H.

k=0

The material covered in this book is for the most part linearly ordered, i.e. each chapter uses results established in preceding ones. The notable exception to this rule are  Chap. 6 devoted to the theory of the trace and  Chap. 7 dealing with functional calculus for families of self-adjoint operators and for normal operators. The results of these chapters are not used in the remaining parts of the book. Also the results of  Chap. 10 on spectral theory of unbounded operators are not used until  Chap. 12. As we already mentioned, the Appendix contains additional material needed in various places in the book arranged into five sections. I wish to thank my teacher professor S.L. Woronowicz who introduced me to the theory of operators on Hilbert spaces and has over the years shared with me his knowledge of the subject. I also thank colleagues and students from the Faculty of Physics and from the Department of Mathematics of the University of Warsaw for their support and helpful remarks during the writing of this book. Warsaw July 2018

Piotr Mikołaj Sołtan

xi

Contents Part I

Bounded Operators

1

Spectrum of an Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1 1.2 1.3

C*-Algebra of Operators on a Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . Spectrum and Spectral Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectrum in C*-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4 11

2

Continuous Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

3

Positive Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.1 3.2 3.3 3.4 3.5

Positive Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Partial Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polar Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Monotone Convergence of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19 24 26 27 29

4

Spectral Theorems and Functional Calculus .. . . . . . . . . . . . . . . . . .

4.1 4.2 4.3 4.4 4.5 4.6

Multiplication Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Borel Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holomorphic Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorems of Fuglede and Putnam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional Calculus in C*-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1 5.2

Compact Operators on a Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fredholm Alternative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

The Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1 6.2 6.3

Definition of the Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trace Class and Hilbert-Schmidt Operators . . . . . . . . . . . . . . . . . . . . . . . . . Hilbert-Schmidt Operators on L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Functional Calculus for Families of Operators . . . . . . . . . . . . . . . . .

7.1 7.2 7.3 7.4

Holomorphic Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joint Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional Calculus for Normal Operators . . . . . . . . . . . . . . . . . . . . . . . . . .

33 34 40 43 47 53 55 59 59 62 69 69 71 84 87 87 91 92 93

xii

Part II

Contents

Unbounded Operators

8

Operators and Their Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.1 8.2 8.3 8.4

Basics of Unbounded Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adjoint Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 105 107 110

z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 113 115 123

10

Spectral Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10.1 10.2 10.3

Continuous Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Borel Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129 129 134 137

9 9.1 9.2 9.3

The Operator T ∗ T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . z-Transform of a Closed Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polar Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

Self-Adjoint Extensions of Symmetric Operators . . . . . . . . . . . . . .

11.1 11.2 11.3

Containment of Operators in Terms of z-Transforms . . . . . . . . . . . . . . . . Cayley Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Krein and Friedrichs Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

One-Parameter Groups of Unitary Operators . . . . . . . . . . . . . . . . . .

12.1 12.2

Stone’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trotter Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A.1 Banach-Steinhaus Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Dynkin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Tensor Product of Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Open Mapping Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Quotient Spaces and Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 147 154 165 166 170 175 175 177 178 181 184 193 195 197

Part I

Bounded Operators

3

Spectrum of an Operator © Springer Nature Switzerland AG 2018 P. Sołtan, A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-319-92061-0_1

1.1

C*-Algebra of Operators on a Hilbert Space

Let H be a Hilbert space. The Banach space B(H) of bounded operators on H is an algebra over C with multiplication defined as composition of operators and the identity operator 1 : H → H is the unit (neutral element of multiplication) of this algebra. The operation of passing to the adjoint operator B(H)  x −→ x ∗ ∈ B(H) is an anti-linear, anti-multiplicative involution (for any x ∈ B(H) we have x ∗∗ = x). Moreover, the operator norm is compatible with algebra structure in the sense that x, y ∈ B(H).

xy ≤ x

y ,

In particular B(H) is a Banach algebra. Proposition 1.1 For x ∈ B(H) we have (1) x = x ∗ , (2) x ∗ x = x 2 . Proof Both equalities are obvious for x = 0. Therefore let us assume that x > 0. Then clearly

x ∗ x ≤ x ∗

x . Furthermore, the computation  

x ∗ x = sup x ∗ xξ = sup sup  η x ∗ xξ 

ξ =1

ξ =1 η =1

  ≥ sup  ξ x ∗ xξ  = sup xξ 2 = x 2 .

ξ =1

ξ =1

1

4

Chapter 1 • Spectrum of an Operator

1 yields

x 2 ≤ x ∗ x ≤ x ∗

x .

(1.1)

Dividing both sides by x we obtain

x ≤ x ∗ , which by symmetry shows that x ∗

x ∗ x = x 2 .

=

x . Substituting this into (1.1) gives 

ⓘ Remark 1.2 Let us note that the proof of Proposition 1.1 can easily be extended to the case when x is an operator between different Hilbert spaces. Thus if H and K are Hilbert spaces then for any x ∈ B(H, K) we have x ∗ x = x 2 .

Proposition 1.1(1) says that the involution x → x ∗ on B(H) is isometric. A Banach algebra together with an isometric, anti-linear and anti-multiplicative involution is called a Banach ∗-algebra, while a Banach ∗-algebra whose involution additionally possesses property (2) from Proposition 1.1 is called a C∗ -algebra. Thus B(H) is a C∗ -algebra. Moreover any norm-closed ∗-subalgebra1 A ⊂ B(H) is also a C∗ -algebra. For any subset S ⊂ B(H) there exists the smallest C∗ -algebra A ⊂ B(H) containing S and it is called the C∗ -algebra generated by S. We denote it by the symbol C∗ (S). It is easy to show that C∗ (S) is the closure of the set of linear combinations of all products of elements of the sets S and S ∗ = {s ∗ s ∈ S}. For x ∈ B(H) we will write C∗ (x) and C∗ (x, 1) instead of C∗ ({x}) and C∗ ({x, 1}). Another example of a C∗ -algebra is the space C(X) of continuous functions on a compact space X with uniform norm · ∞ , pointwise addition and multiplication and involution given by f → f . A seemingly different example would be the space Cb (Y ) of bounded continuous functions on a locally compact space Y with norm · ∞ and pointwise algebraic operations. However this example is not really different from the previous one, as in fact Cb (Y ) is naturally isomorphic to C(βY ), where βY is the Stoneˇ Cech compactification of Y .2

1.2

Spectrum and Spectral Radius

Let x ∈ B(H). Recall that x is invertible if there exists an operator y ∈ B(H) such that xy = yx = 1. The resolvent set of x is   ρ(x) = λ ∈ C the operator λ1 − x is invertible , and its complement σ (x) = C K ρ(x) is called the spectrum of x. 1 Vector 2 Cf.

subspace closed under composition of operators and passing to the adjoint operator. [Eng, Corollary 3.6.3].

1

5 1.2 · Spectrum and Spectral Radius

It is known that the set of invertible operators is open (see below for an argument proving this) and since ρ(x) is the pre-image of this set under the continuous map C  λ −→ λ1 − x ∈ B(H), we see that the resolvent set is open. Moreover, if λ0 ∈ ρ(x) and λ ∈ C satisfies |λ − λ0 | <

1 ,

(λ0 1−x)−1

then λ ∈ ρ(x) and it is easy to see that (λ1 − x)−1 =

∞  (λ0 − λ)n (λ0 1 − x)−n−1 . n=0

In particular the mapping ρ(x)  λ → (λ1 − x)−1 ∈ B(H) called the resolvent of x is holomorphic. ⓘ Remark 1.3 For any λ, μ ∈ ρ(x) we have (λ1 − x)−1 − (μ1 − x)−1 = (μ − λ)(λ1 − x)−1 (μ1 − x)−1 .

(1.2)

(in particular the values of the resolvent of x at different points of ρ(x) commute). Indeed: the formula is obvious for λ = μ and for λ = μ we easily check that 1 μ−λ

 (λ1 − x)−1 − (μ1 − x)−1 (μ1 − x)(λ1 − x)  1 = μ−λ (λ1 − x)−1 (μ1 − x) − 1 (λ1 − x)   1 = μ−λ (λ1 − x)−1 (μ − λ)1 + (λ1 − x) − 1 (λ1 − x)  1 = μ−λ (μ − λ)(λ1 − x)−1 + 1 − 1 (λ1 − x) = 1

and  1 (μ1 − x)(λ1 − x) μ−λ (λ1 − x)−1 − (μ1 − x)−1  1 = μ−λ (μ1 − x) 1 − (λ1 − x)(μ1 − x)−1   1 = μ−λ (μ1 − x) 1 − (λ − μ)1 + (μ1 − x) (μ1 − x)−1  1 = μ−λ (μ1 − x) 1 − (λ − μ)(μ1 − x)−1 + 1 = 1. It follows that (μ1 − x)(λ1 − x) is invertible and its inverse is the operator  1 −1 − (μ1 − x)−1 . Formula (1.2) is known as the resolvent identity or μ−λ (λ1 − x) the resolvent formula.

6

Chapter 1 • Spectrum of an Operator

1 Proposition 1.4 Let x, y ∈ B(H). Then σ (xy) ∪ {0} = σ (yx) ∪ {0}.

(1.3)

Proof Let λ ∈ ρ(yx) K {0}, so that the operator λ1 − yx is invertible. We have (λ1 − xy)

 1 λ

1 + x(λ1 − yx)−1 y



 λ1 − xy + (λ1 − xy)x(λ1 − yx)−1 y  = λ1 λ1 − xy + x(λ1 − yx)(λ1 − yx)−1 y =

1 λ

= λ1 (λ1 − xy + xy) = 1 and

 1 −1 λ 1 + x(λ1 − yx) y (λ1 − xy)  λ1 − xy + x(λ1 − yx)−1 y(λ1 − xy)  = λ1 λ1 − xy + x(λ1 − yx)−1 (λ1 − yx)y =

1 λ

= λ1 (λ1 − xy + xy) = 1  which means that λ1 − xy is invertible with inverse λ1 1 + x(λ1 − yx)−1 y . In other words λ ∈ ρ(xy) K {0} and this shows that ρ(yx) K {0} ⊂ ρ(xy) K {0}. By symmetry, also ρ(xy) K {0} ⊂ ρ(yx) K {0} and hence ρ(xy) K {0} = ρ(yx) K {0}. In other words σ (xy) ∪ {0} = σ (yx) ∪ {0}. 

ⓘ Remark 1.5 By taking set difference of both sides of (1.3) with {0} we get another useful expression of the same phenomenon: σ (xy) K {0} = σ (yx) K {0}.   Proposition 1.6 For any x ∈ B(H) we have λ ∈ C |λ| > x ⊂ ρ(x). Proof If |λ| > x then the series ∞  n=0

λ−n−1 x n =

1 λ

∞ 

λ−n x n

n=0

converges in B(H), because λ−n x n ≤ of λ1 − x.



x

|λ|

n

. One easily checks that its sum is the inverse 

1

7 1.2 · Spectrum and Spectral Radius

Rephrasing Proposition 1.6, for any x ∈ B(H) we have     σ (x) = C K ρ(x) ⊂ C K λ ∈ C |λ| > x = λ ∈ C |λ| ≤ x , and therefore the spectrum of x is a closed and bounded (hence compact) subset of C. Theorem 1.7 For any x ∈ B(H) we have (1) σ (x) is non-empty,  1 (2) the sequence x n n n∈N is convergent and   1 lim x n n = sup |λ| λ ∈ σ (x) .

n→∞

Proof Define 1

α(x) = inf x n n ,

x ∈ B(H).

n∈N

1

Take any ε > 0. Then there is a natural number nε ∈ N such that x nε nε ≤ α(x) + ε or, in other words,  n

x nε ≤ α(x) + ε ε . Take now any n ∈ N and divide it by nε with remainder, i.e. find q, r ∈ Z+ such that n = qnε + r and r < nε . Then we have  qn  n−r

x r .

x n = x qnε x r ≤ x nε q x r ≤ α(x) + ε ε x r = α(x) + ε Therefore  1− r 1 r n x n

x n n ≤ α(x) + ε which shows that 1

1

α(x) ≤ lim inf x n n ≤ lim sup x n n ≤ α(x) + ε. n→∞

n→∞

 1 As ε is arbitrary, we find that the sequence x n n n∈N is convergent (with limit α(x)).

8

Chapter 1 • Spectrum of an Operator

1 In particular, if x, y ∈ B(H) commute then 1 1 α(xy) = lim (xy)n n = lim x n y n n n→∞

n→∞

1 n

1 n

1

1

≤ lim x y = lim x n n lim x n n = α(x)α(y). n

n

n→∞

n→∞

Now if |λ| > α(x) then the series



n=0

∞ 

n→∞

x n

|λ|n

converges, and therefore so does

λ−n x n .

n=0

We easily check that its sum is the inverse of 1 − xλ . It follows that λ1 − x is invertible and we obtain   α(x) ≥ sup |λ| λ ∈ σ (x) .

(1.4)

For the proof of the reverse inequality and non-emptiness of the spectrum we need to consider two cases.

Case 1 α(x) = 0. In this case x is not invertible (and so 0 ∈ σ (x)). Indeed: if x were invertible we would have 1 = α(1) = α(xx −1 ) ≤ α(x)α(x −1 ) = 0. 



Case 2 α(x) > 0. Let us assume that α(x) > sup |λ| λ ∈ σ (x) . Since σ (x) is a compact subset of C, there exists r ∈ ]0, α(x)[ with the property that   σ (x) ⊂ λ ∈ C |λ| ≤ r .   Thus the set D = λ ∈ C |λ| > r is contained in ρ(x). For any continuous functional ϕ on B(H) the function  D  λ −→ ϕ (λ1 − x)−1 ∈ C is holomorphic. Moreover, for |λ| > α(x) we have ∞   ϕ (λ1 − x)−1 = λ−n−1 ϕ(x n ). n=0

9 1.2 · Spectrum and Spectral Radius

This function vanishes at infinity,3 so ⎧ ⎨0, f (μ) = ⎩ϕ  1 1 − x −1 , μ

μ = 0, 0 < |μ| <

1 r

  defines a holomorphic function on D −1 = μ ∈ C |μ| < 1r with Taylor expansion around zero given by ∞ 

μn+1 ϕ(x n ).

n=0

This expansion must converge on the whole disk D −1 and, in particular, for any μ ∈ D −1 we have lim μn+1 ϕ(x n ) = 0.

n→∞

−1 and Take now λ0 such that r < |λ0 | < α(x). Then λ−1 0 ∈D

lim λ−n−1 ϕ(x n ) n→∞ 0

= 0.

Consider the family (sequence) of continuous functionals on B(H)∗ given by ϕ(x n ) ∈ C, B(H)∗  ϕ −→ λ−n−1 0

n ∈ N.

By the Banach-Steinhaus theorem there exists a constant M < +∞ such that sup |λ0 |−n−1 x n ≤ M. n∈N

Thus

x n ≤ M|λ0 |n+1 ,

n∈N

and consequently 1

1

1

α(x) = lim x n n ≤ lim M n |λ0 |1+ n = |λ0 | < α(x). n→∞

3 For

n→∞

|μ| > 1 we have    ∞   (λμ)−n−1 ϕ(x n ) ≤  n=0

ϕ

|μ|

∞ 

x n

|λ|n+1

|μ|−n ≤

n=0

ϕ

|μ|

∞  n=0

1

x n

−−−−→ |λ|n+1 |μ|→∞

0.

The series is convergent because |λ| > lim x n n , i.e. |λ|−1 is strictly smaller than the radius of convergence n→∞  1 −1 R = lim sup x n n . n→∞

1

10

Chapter 1 • Spectrum of an Operator

1   This contradiction shows that it is not possible to have α(x) > sup |λ| λ ∈ σ (x) . In other words   α(x) ≤ sup |λ| λ ∈ σ (x) ,   which together with (1.4) gives α(x) = sup |λ| λ ∈ σ (x) .



For any x ∈ B(H) the quantity   sup |λ| λ ∈ σ (x) is called the spectral radius of x and is denoted by |σ (x)|. Here are a few properties of the spectral radius: ▬ for any x we have |σ (x)| ≤ x , ▬ if x, y ∈ B(H) commute then |σ (xy)| ≤ |σ (x)||σ (y)|, ▬ |σ (x ∗ )| = |σ (x)|. The last property follows immediately from the fact that   σ (x ∗ ) = σ (x) = λ λ ∈ σ (x) . Proposition 1.8 Let x ∈ B(H) and let p(λ) = α0 + α1 λ + · · · + αn λn be a polynomial. Define p(x) = α0 1 + α1 x + · · · + αn x n . Then     σ p(x) = p σ (x) = p(λ) λ ∈ σ (x) . Proof The statement is obvious for n = 0. Assume therefore that n ≥ 1. Take λ0 ∈ σ (x), so that λ0 1 − x is not invertible. Then the operator p(λ0 )1 − p(x) cannot be invertible, as p(λ0 )1 − p(x) =

n 

αk (λk0 1 − x k )

k=0

=

n 

αk (λk0 1 − x k ) = (λ0 1 − x)

k=1

and λ0 1 − x and

n

k=1

αk

k−1

j =1

n  k=1

k−j j −1 x

λ0

commute.

αk

k−1  j =1

k−j j −1

λ0

x

.

1

11 1.3 · Spectrum in C*-Algebras

This shows that   p σ (x) ⊂ σ p(x) .   On the other hand, if μ ∈ p(λ) λ ∈ σ (x) and λ1 , . . . , λn are the zeros of the polynomial μ − p(λ) then clearly λ1 , . . . , λn ∈ σ (x). Furthermore μ − p(λ) = γ (λ1 − λ)m1 · · · (λn − λ)mn for some m1 , . . . , mn ∈ N and γ = 0, and therefore μ1 − p(x) = γ (λ1 1 − x)m1 · · · (λn 1 − x)mn . Thus μ1 − p(x) is invertible (as a product of invertible operators) and this means that μ ∈  σ p(x) . We have therefore shown that   p σ (x) ⊃ σ p(x) . 

1.3

Spectrum in C*-Algebras

Let A be a C∗ -algebra with unit (examples being B(H) for a Hilbert space H or C(X) for a compact space X, but these are far from exhaustive) and let a ∈ A. Just as for elements of B(H) we say that a is invertible if there exists b ∈ A such that ab = ba = 1. We go on to define the spectrum of a:   σ (a) = λ ∈ C the element λ1 − a is not invertible . Consider for example the C∗ -algebra A = C(X) with X a compact space. Then it is easy to see that the spectrum of an element f ∈ A coincides with the range of the function f . Theorem 1.9 Let a be an element of a C∗ -algebra with unit. Then (1) σ (a) is a non-empty compact subset of C contained in the disk with center 0 and radius a ,  1 (2) the sequence a n n n∈N is convergent and   1 lim a n n = sup |λ| λ ∈ σ (a) ,

n→∞

(Continued )

12

Chapter 1 • Spectrum of an Operator

1 Theorem 1.9 (continued) (3) for any b ∈ A we have σ (ab) ∪ {0} = σ (ba) ∪ {0},   (4) for any polynomial p ∈ C[ · ] we have σ p(a) = p σ (a) .

The proofs of all statements in the above theorem can be carried out by substituting A for B(H) in the proofs of Theorem 1.7 and Propositions 1.4 and 1.8.

Notes Basics of the theory of operators on Hilbert spaces recalled in this chapter can be found in every textbook on the subject, including almost all items from the bibliography listed at the end of this book. More information on spectra of operators and elements of C∗ algebras can be found e.g. in [Arv2 , Chapters 1 and 2], [Ped, Chapter 4], [Mau, Chapter VIII] as well as in many other books. A natural generalization of operator theory is provided by the theory of Banach algebras and in particular of C∗ -algebras ([Arv1 , Zel]). Within this approach many results can be obtained in a very elegant way, the price for this being paid by way of making the presentation more abstract and using a number of complicated structures. In this book, however, we aim to minimize the use of abstract theory reserving the more sophisticated path for further investigations on the part of the reader. A wealth of great examples and exercises on the subject of spectra can be found in [Hal, Sections 9, 10 and 11] and in problem sections of textbooks such as [Arv1 , Arv2 , ReSi1 , Rud2 ].

2

13

Continuous Functional Calculus © Springer Nature Switzerland AG 2018 P. Sołtan, A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-319-92061-0_2

In this chapter we will introduce by far the most important tool of the theory of operators on Hilbert space, namely functional calculus for self-adjoint operators. We begin with slightly more general considerations focused on normal operators which we will revisit later in  Chap. 7. An operator x ∈ B(H) is called normal if x ∗ x = xx ∗ . Equivalently x is normal if and only if for any ξ ∈ H we have xξ = x ∗ ξ . Indeed: if x ∗ x = xx ∗ then      

xξ 2 = xξ xξ  = ξ x ∗ xξ = ξ xx ∗ξ = x ∗ ξ x ∗ ξ = x ∗ ξ 2 . Conversely if xξ 2 = x ∗ ξ 2 for all ξ ∈ H then the sesquilinear forms   (ξ, η) −→ ξ x ∗ xη

and

  (ξ, η) −→ ξ xx ∗η

coincide on pairs of vectors of the form (ξ, ξ ), so by the polarization identity they are equal. Consequently x ∗ x = xx ∗. An operator x ∈ B(H) is self-adjoint when x = x ∗ . Clearly, a self-adjoint operator is normal. Also any unitary operator, i.e. u ∈ B(H) satisfying u∗ u = uu∗ = 1, is normal. Proposition 2.1 Let x ∈ B(H) be a normal operator and let λ ∈ C and ψ ∈ H be such that xψ = λψ.1 Then x ∗ ψ = λψ. Proof One easily checks that the operator λ1 − x is normal, and so (λ1 − x ∗ )ψ = (λ1 − x)∗ ψ = (λ1 − x)ψ = 0. 

1 Restriction

of a compact operator to an invariant subspace clearly is compact.

14

2

Chapter 2 • Continuous Functional Calculus

Corollary 2.2 Let x ∈ B(H) be a normal operator and let λ, μ ∈ σ (x) be different eigenvalues of x. Then the eigenspaces of x for λ and μ are orthogonal. Proof Let ψ and ϕ be eigenvectors of x for the eigenvalues λ and μ respectively: xψ = λψ and xϕ = μϕ. Using Proposition 2.1 we compute     μ ψ ϕ = ψ xϕ = x ∗ ψ ϕ = λψ ϕ = λ ψ ϕ , which shows that (λ − μ) ψ ϕ = 0. As λ = μ, we must have ψ ϕ = 0.



In particular eigenspaces of a self-adjoint operator for different eigenvalues also have to be orthogonal. Proposition 2.3 Let x ∈ B(H). Then (1) if x is normal then |σ (x)| = x , (2) if x is self-adjoint then σ (x) ⊂ R. Proof Assume first that x is normal. We have

x 2 2 = (x 2 )∗ (x 2 ) = x ∗ x ∗ xx

= x ∗ xx ∗ x = (x ∗ x)∗ (x ∗ x) = x ∗ x 2 = x 4 , k

so that x 2 = x 2 . Since for any k ∈ N the operator x 2 is normal as well, for any n ∈ N we have n−1 n n−1 n−2 n

x 2 = (x 2 )2 = x 2 2 = x 2 4 = · · · = x 2 .  1 In particular, the sequence x n n n∈N has a subsequence which is constant and equal to

x . On the other hand 1

|σ (x)| = lim x n n . n→∞

Now let x be self-adjoint. Take any λ ∈ σ (x) and write λ = α + iβ with α, β ∈ R. For n ∈ N consider the operator xn = x − (α − inβ)1. We have i(n + 1)β ∈ σ (xn ), so that (n + 1)2 β 2 ≤ |σ (xn )|2 ≤ xn 2 = xn ∗ xn

= (x − (α + inβ)1 (x − (α − inβ)1 = (x − α1)2 + n2 β 2 1 ≤ (x − α1)2 + n2 β 2 for all n. This means that β = 0 and consequently λ ∈ R.



15 Chapter 2 • Continuous Functional Calculus

Before stating the main theorem of this chapter let us recall that a ∗-isomorphism is a bijective map between algebras with involution preserving addition, multiplication and involution. Theorem 2.4 (Continuous Functional Calculus)  Let x ∈ B(H) be self-adjoint. Then there exists a unique map C σ (x) → B(H) denoted by  C σ (x)  f −→ f (x) ∈ B(H) such that ▬ if f is a polynomial function f (λ) = α0 + α1 λ + · · · + αn λn then f (x) = α0 1 + α1 x + · · · + αn x n ,  ▬ f (x) = f ∞ for all f ∈ C σ (x) .  Moreover the map f → f (x) is a ∗-isomorphism of the C∗ -algebra C σ (x) onto C∗ (x, 1).

Proof  Let P σ (x) be the set of polynomial functions on σ (x),  i.e. restrictions of polynomials to  σ (x) and let  : C[ · ] → C σ (x) be the map p → pσ (x) . For a polynomial p ∈ C[ · ] we have       p(x) = σ p(x)  = sup |μ| μ ∈ σ p(x)   = sup |p(λ)| λ ∈ σ (x) = (p) ∞ . Thus the mapping C[ · ]  p −→ p(x) ∈ B(H)  factorizes through , i.e. there exists a linear  : P σ (x) → B(H) such that  p(x) =  (p) . Moreover  is an isometry.   The space P σ (x) is dense in C σ (x) and its image under  is dense in the C∗ -algebra  C∗ (x, 1). Therefore  extends uniquely to an isometry from C σ (x) onto C∗ (x, 1) which we will from now on denote by f → f (x). It is clear that this map satisfies

2

16

2

Chapter 2 • Continuous Functional Calculus

▬ if f is a polynomial function f (λ) = α0 + α1 λ + · · · + αn λn then f (x) = α0 1 + α1 x + · · · + αn x n ,  ▬ f (x) = f ∞ for all f ∈ C σ (x) , and that these conditions determine this map uniquely. The properties (f + g)(x) = f (x) + g(x), (fg)(x) = f (x)g(x), (λf )(x) = λf (x), f (x) = f (x)∗ ,

 f, g ∈ C σ (x) ,  f, g ∈ C σ (x) ,  λ ∈ C, f ∈ C σ (x) ,  f ∈ C σ (x)

 are easily checked on elements of P σ (x) and for general continuous functions on σ (x) we use approximation by polynomial functions. In other words f → f (x) is an isometric  ∗-isomorphism of C σ (x) onto C∗ (x, 1). 

ⓘ Remark 2.5 (1) Let x ∈ B(H) be self-adjoint and let f be a continuous function on σ (x) which can be uniformly approximated by polynomials without constant term. Then f (x) belongs to C∗ (x) which can be strictly smaller than C∗ (x, 1).  (2) If 0 ∈ σ (x) (i.e. x is invertible) then any f ∈ C σ (x) can be uniformly approximated by polynomials without constant term. In particular if x is an invertible self-adjoint operator then the constant function 1 on σ (x) can be uniformly approximated by polynomials without constant term. Therefore 1 belongs to C∗ (x) and consequently C∗ (x, 1) = C∗ (x). Moreover, also the function ı : σ (x)  λ −→ λ−1 ∈ C can be uniformly approximated by polynomials without constant term. It is easy to see that ı(x) = x −1 , so x −1 ∈ C∗ (x). (3) Even when x is not self-adjoint, but 0 ∈ σ (x) we can show that x −1 ∈ C∗ (x). Indeed: the operator x ∗ x is self-adjoint and invertible with inverse x −1 (x −1 )∗ . It follows from (2) that x −1 (x −1 )∗ belongs to C∗ (x ∗ x) ⊂ C∗ (x). Therefore  x −1 = x −1 (x −1 )∗ x ∗ ∈ C∗ (x). Proposition 2.6 Let x and y be commuting self-adjoint operators on H. Then for any f ∈   C σ (x) and g ∈ C σ (y) we have f (x)g(y) = g(y)f (x).

2

17 Chapter 2 • Continuous Functional Calculus

Proof Let (fn )n∈N and (gn )n∈N be sequences of polynomials uniformly approximating f and g on σ (x) and σ (y) respectively. It is clear that for any n we have fn (x)gn (y) = gn (y)fn (x) and fn (x)gn (y) −−−→ f (x)g(y) n→∞

and

gn (y)fn (x) −−−→ g(y)f (x). n→∞



ⓘ Remark 2.7 Let x, y ∈ B(H) and assume the x = x ∗ and xy = yx. Then for any

 f ∈ C σ (x) the operator f (x) also commutes with y. The proof follows the lines of the proof of Proposition 2.6 .

Theorem 2.8 (Spectral Mapping Theorem)  Let x ∈ B(H) be self-adjoint. Then for any f ∈ C σ (x) we have   σ f (x) = f σ (x) .

Proof For any μ μ1 − f (x) = (μ − f )(x). Therefore μ1 − f (x) is invertible if and only if the function μ − f is an invertible element   of C σ (x) , i.e. is and only if μ ∈ f σ (x) . 

ⓘ Remark 2.9 Let x be a self-adjoint operator and assume that x n = 0 for some n ∈ N.   Then σ (x) = {0}, because σ (x n ) = {0} and σ (x n ) = λn λ ∈ σ (x) . It follows that

x = |σ (x)| = 0, i.e. x = 0.

 Proposition 2.10 Let x ∈ B(H) be self-adjoint. Then for any real-valued g ∈ C σ (x) the   operator g(x) is self-adjoint and for any f ∈ C g σ (x) we have  f g(x) = (f ◦ g)(x). Proof  Since C σ (x)  g → g(x) ∈ C∗ (x, 1) is a ∗-isomorphism, g = g implies g(x)∗ = g(x).  So let us fix a real-valued g ∈ C σ (x) and consider the mapping   C g σ (x)  f −→ (f ◦ g)(x) ∈ B(H).

18

2

Chapter 2 • Continuous Functional Calculus

It is easy to see that it is isometric and maps polynomial functions onto corresponding polynomials in g(x). By uniqueness of the continuous functional calculus it must be equal to  the map f → f g(x) . 

Notes It is difficult to find a more fundamental concept in the theory of operators on Hilbert spaces than functional calculus and, in particular, the continuous functional calculus. As we already mentioned earlier, it is the first step in the development of the theory of C∗ -algebras and general operator algebras. The reader will find plentiful exercises and examples e.g. in [Arv1 , Chapter 1], [Arv2 , Chapter 2], [Hal, Section 15], [Ped, Sections 4.4 and 4.5], [ReSi1 , Chapter VII], [Rud2 , Chapters 10 and 12]. Let us also note that continuous functional calculus for self-adjoint operators can be extended to a larger class of operators (namely the so called normal operators, see  Sect. 4.6). We will do this in  Sect. 7.4.

19

Positive Operators © Springer Nature Switzerland AG 2018 P. Sołtan, A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-319-92061-0_3

One of the first and, incidentally, very effective applications of functional calculus is to take square roots of positive operators. In this chapter we will introduce positive operators and related notions of a projection and a partial isometry. We will prove existence and uniqueness of polar decomposition of bounded operators. We will also briefly investigate the partial order on operators defined by the notion of a positive operator. In particular we will show that any bounded monotonically increasing net of self-adjoint operators has a supremum which is also the limit of the net in strong topology.

3.1

Positive Operators

An operator x ∈ B(H) is called positive if x is self-adjoint and σ (x) ⊂ R+ . In this case we write x ≥ 0. The set of positive operators on H will be denoted by the symbol B(H)+ . Theorem 3.1 Let x ∈ B(H) be positive. Then there exists a unique positive a ∈ B(H) such that a 2 = x.

Proof 1 The function f : λ → λ 2 is continuous on σ (x). Put a = f (x). Clearly a 2 = f (x)f (x) = x,  since f (λ)f (λ) = λ for all λ ∈ σ (x). Moreover σ (a) = f σ (x) is contained in R+ , so that a ∈ B(H)+ .

3

20

Chapter 3 • Positive Operators

Now let b be a positive operator such that b2 = x. By Proposition 2.10 we have b =  f g(b) , where g(λ) = λ2 for all λ ∈ R+ . Therefore

3

b = f (b2 ) = f (x) = a. 

Let x ∈ B(H) be a positive operator. The unique positive operator a such that a 2 = x is called the square root of x. It follows from the existence of square roots that any positive operator is the square of a self-adjoint operator. Moreover, by the spectral mapping theorem, the square of a self-adjoint operator y is positive, since its spectrum consists of squares of elements of σ (y). Proposition 3.2 Let x ∈ B(H) be self-adjoint. Then there exists a unique pair (a, b) of positive operators such that x =a−b

ab = 0.

and

(3.1)

Proof Consider the following two continuous functions on σ (x): ⎧ ⎨λ, f (λ) = ⎩0,

λ ≥ 0,

and

λ 0,

⎩−λ,

λ ≤ 0.

It is check that a = f (x) and b = g(x) satisfy the conditions (3.1). On the other hand if a pair (a, b) of positive operators satisfies (3.1) then first of all a and b must commute: ba = b∗ a ∗ = (ab)∗ = 0∗ = 0 = ab. 1

1

1

1

It follows that also a 2 and b 2 commute (Proposition 2.6) and, moreover, a 2 b 2 = 0, since  1 1 2 a 2 b 2 = ab = 0 (cf. Remark 2.9). From this we obtain  1 1 2 a 2 + b 2 = a + b, so a + b is positive, as a square of a self-adjoint operator. Furthermore (a + b)2 = (a − b)2 = x 2 ,

3

21 3.1 · Positive Operators

1

which means that a+b is the unique square root of the positive operator x 2 , i.e. a+b = (x 2 ) 2 . Consequently a=

1 2



 1 (a + b) + (a − b) = 12 (x 2 ) 2 + x = f (x)

and similarly b=

1 2

  1 (a + b) − (a − b) = 12 (x 2 ) 2 − x = g(x). 

Operators a and b defined in Proposition 3.2 are called the positive part and negative part of the self-adjoint operator x. We denote them by the symbols x + and x − respectively. Note that x + and x − belong to C∗ (x). Proposition 3.3 Let x ∈ B(H). Then (1) x = 0 if and only if for any ξ ∈ H we have ξ xξ  = 0, (2) x = x ∗ if and only if for any ξ ∈ H we have ξ xξ  ∈ R. Proof Ad (1). It follows from the polarization formula that if ξ xξ  = 0 for all ξ then the sesquilinear form (ξ, η) −→ ξ xη ,

ξ, η ∈ H

is the zero form, and so x = 0. Ad (2). Consider two sesquilinear forms: F1 : (ξ, η) −→ ξ xη

and

F2 : (ξ, η) −→ xξ η .

We have F1 (ξ, ξ ) = ξ xξ  = ξ xξ  = xξ ξ  = F2 (ξ, ξ ), because ξ xξ  ∈ R. It follows that F1 = F2 , so that x is self-adjoint. Proposition 3.4 Let x ∈ B(H). Then the following conditions are equivalent: (1) x is positive, (2) there exists a self-adjoint operator y such that x = y 2 , (3) there exists an operator z such that x = z∗ z, (4) for any ξ ∈ H we have ξ xξ  ≥ 0.



22

3

Chapter 3 • Positive Operators

Proof 1 (1) ⇒ (2) follows from Theorem 3.1: y = x 2 . The implication (2) ⇒ (3) is obvious and so is (3) ⇒ (4): for any ξ we have   ξ xξ  = ξ z∗ zξ = zξ zξ  = zξ 2 ≥ 0. (4) ⇒ (1). Decompose x into its positive and negative part, x = x + − x − , and let 3 f (λ) = λ 2 for λ ∈ R+ . Since x + x − = 0, for any ξ we have        0 ≤ x − ξ x(x − ξ ) = ξ (x − xx − )ξ ) = ξ x − (x + − x − )x − ξ )     = − ξ (x − )3 ξ = − x − ξ x − (x − ξ )   1 1 = − x − ξ (x − ) 2 (x − ) 2 (x − ξ )   = − f (x − )ξ f (x − )ξ ≤ 0.   It follows that ξ (x − )3 ξ = 0 for all ξ , which by Proposition 3.3(1) means that (x − )3 = 0. It follows now from Remark 2.9 that x − = 0. In particular x = x + ≥ 0.



In the terminology of quantum physics, for x ∈ B(H) the quantity ξ xξ  is referred to as the expectation value of x in the state determined by ξ (albeit this terminology usually applies only to vectors ξ of norm 1). Proposition 3.4 says, among other things, that an operator is positive if and only if its expectation values in all states are positive. The notion of positivity for elements of B(H) allows us to define a partial order relation of B(H): let x, y ∈ B(H). We say that x dominates y if x − y ≥ 0. We write this as x ≥ y. Let us note for future reference that for any x ∈ B(H) we have x ∗ x ≤ x 2 1, The proof is based on a simple computation: 

   ξ x ∗ xξ = xξ 2 ≤ x 2 ξ 2 = ξ x 2 ξ ,

ξ ∈ H.

Moreover, if x ∗ x ≤ 1, then by the same computation x ≤ 1. In other words we have the following: Proposition 3.5 For any positive t ∈ B(H)



t ≤ 1 ⇐⇒ t ≤ 1 .

Proposition 3.4 can be used to obtain information about the partial order on B(H) both of geometric and algebraic nature: Corollary 3.6 B(H)+ is a convex cone, i.e. (1) if x ∈ B(H)+ and λ ∈ R+ then λx ∈ B(H)+ (2) if x, y ∈ B(H)+ then x + y ∈ B(H)+ ,  (3) B(H)+ ∩ − B(H)+ = {0}. Moreover, if x ∈ B(H)+ and y ∈ B(H) then y ∗ xy ∈ B(H)+ .

3

23 3.1 · Positive Operators

Proof The first of the listed properties of B(H)+ follows immediately from the definitions, while the second is a consequence of characterization of positivity given by point (4) of Proposition 3.4.  If x ∈ B(H)+ ∩ − B(H)+ then σ (x) = {0}, and since x is self-adjoint, we get x = 0. Let x ∈ B(H)+ and y ∈ B(H). Then for any ξ ∈ H we have 

 ξ y ∗ xyξ = yξ x(yξ ) ≥ 0,

which means that y ∗ xy ≥ 0.



Let us note here that functional calculus for self-adjoint operators is compatible with the order structure on B(H) in the sense that if x ∈ B(H) is self-adjoint and f, g ∈ C σ (x) satisfy f ≥ g then f (x) ≥ g(x) in B(H) because ∗   1 1 f (x) − g(x) = (f − g)(x) = (f − g) 2 (x) (f − g) 2 (x) ∈ B(H)+ . Proposition 3.7 Let x, y ∈ B(H)+ be such that 0 ≤ x ≤ y and assume that x is invertible. Then y is invertible and y −1 ≤ x −1 . Proof The compact set σ (x) lies in [0, +∞[ and it does not contain 0. Therefore there exists δ > 0 such that σ (x) ⊂ [δ, +∞[ . By functional calculus we immediately get x ≥ δ1, and so y ≥ δ1. Thus, again by functional calculus, we find that y is invertible. 1 Now multiplying both sides of the inequality y ≥ x by y − 2 from left and right and using Corollary 3.6 we obtain 1

1

y − 2 xy − 2 ≤ 1 1 1 which, in particular, means that y − 2 xy − 2 ≤ 1. Thus 1 − 1 2 − 1 − 1 x 2 y 2 = y 2 xy 2 ≤ 1 and consequently − 1 1  1 − 1 ∗ y 2 x 2 = x 2 y 2 ≤ 1. This, in turn, shows that  − 1 1 ∗  − 1 1 − 1 1 2 y 2 x 2 y 2 x 2 ≤ y 2 x 2 1 ≤ 1 or in other words 1

1

x 2 y −1 x 2 ≤ 1.

24

Chapter 3 • Positive Operators

1

Multiplying both sides of this inequality by x − 2 from left and right we arrive at y −1 ≤ x −1 .



3

Taking roots of positive operators has many other applications. An example of such an application is provided by the following useful proposition: Proposition 3.8 Any x ∈ B(H) is a linear combination of four unitary operators. Proof Writing x = 12 (x + x ∗ ) + i 2i1 (x − x ∗ ), we express x as a linear combination of self-adjoint operators. Now any self-adjoint operator y ∈ B(H) of norm 1 can be written in the form y=

1 2



 1 1 y + i(1 − y 2 ) 2 + y − i(1 − y 2 ) 2 . 1

The operators y ± i(1 − y 2 ) 2 are unitary, since  1 ∗  1 y ± i(1 − y 2 ) 2 y ± i(1 − y 2 ) 2  1  1 = y ∓ i(1 − y 2 ) 2 y ± i(1 − y 2 ) 2 = y 2 + (1 − y 2 ) = 1  1  1 ∗ and similarly y ± i(1 − y 2 ) 2 y ± i(1 − y 2 ) 2 = 1.

3.2



Projections

A projection is an operator e ∈ B(H) such that e = e2 and e = e∗ . We are using the term “projection” in a way which is slightly more restrictive than usual. More precisely, the term “projection” is often taken to mean “idempotent operator”, i.e. an x ∈ B(H) such that x 2 = x. We have chosen to use the more restrictive definition requiring that all projection be self-adjoint, because the extra flexibility of working with general idempotents will not have any significance for the topics considered in this book. We leave it to the reader to check that an idempotent operator is self-adjoint if and only if its kernel and range are orthogonal. Any projection e is a positive operator of norm 1 or 0. Moreover the operator  1 − e is also a projection. The set of all projections in B(H) will be denoted by Proj B(H) . Note that an operator e ∈ B(H) is a projection if and only if it is self-adjoint and the two functions λ → λ2 and λ → λ coincide on σ (e). It follows that σ (e) ⊂ {0, 1}. Conversely, if e ∈ B(H) is self adjoint and σ (e) ⊂ {0, 1} then e2 = e precisely because the functions λ → λ2 and λ → λ are equal on σ (e).

3

25 3.2 · Projections

Let e ∈ B(H) be a projection. Then the subspace eH ⊂ H is closed and (1 − e)H is the orthogonal complement of eH. Of course for any closed subspace S ⊂ H there exists a unique projection e ∈ B(H) such that S = eH. It is not hard to check that if e and f are projections then e ≤ f if and only if eH ⊂ f H. We say that two projections e1 , e2 ∈ B(H) are orthogonal if e1 e2 = 0. This is equivalent to the subspaces e1 H and e2 H being orthogonal. Let us also note that for any two projections e and f the operator e + f is a projection if and only if e and f are orthogonal. With any operator x ∈ B(H) we can associate two projections l(x) and r(x): l(x) is defined to be the projection onto the closure of x H, while r(x) is the projection onto the orthogonal complement of ker x. We have l(x)x = x = xr(x). The projections l(x) and r(x) are respectively called the left support and the right support of x. Since for any x ∈ B(H) we have (x H)⊥ = ker x ∗ (η xξ  = 0 for all ξ if and only if x ∗ η ξ  = 0 for all ξ ), it is easy to see that l(x) = r(x ∗ ) (equivalently l(x ∗ ) = r(x) for all x). In particular if x = x ∗ then l(x) = r(x) and, in this case, the common value of l(x) and r(x) is denoted by s(x) and called the support of x. Lemma 3.9 For any x ∈ B(H) we have ker x = ker x ∗ x. Proof It is clear that ker x ⊂ ker x ∗ x. On the other hand, if ξ ∈ ker x ∗ x then ξ x ∗ xξ  = 0, so

xξ 2 = 0.

 Proposition 3.10 For x ∈ B(H) we have r(x) = s(x ∗ x) and l(x) = s(xx ∗ ). Proof Using Lemma 3.9 we see that r(x) = projection onto (ker x)⊥ = projection onto (ker x ∗ x)⊥ = r(x ∗ x) = s(x ∗ x) and similarly l(x) = projection onto x H = projection onto (ker x ∗ )⊥ = projection onto (ker xx ∗ )⊥ = r(xx ∗ ) = s(xx ∗ ). 

3

26

Chapter 3 • Positive Operators

3.3

Partial Isometries

Proposition 3.11 Let v ∈ B(H). Then the following conditions are equivalent: (1) v ∗ v is a projection, (2) vv ∗ v = v, (3) v ∗ vv ∗ = v ∗ , (4) vv ∗ is a projection. Proof (1) and (4) are equivalent by Proposition 1.4 and remarks at the beginning of  Sect. 3.2. More precisely σ (v ∗ v) is contained in {0, 1} if and only if so is σ (vv ∗ ), and furthermore both v ∗ v and vv ∗ are self-adjoint. It follows that v ∗ v is a projection if and only if vv ∗ is a projection. (1) ⇒ (2): the support of any projection e is equal to e. Therefore, setting e = v ∗ v we obtain v ∗ v = s(v ∗ v) = r(v) by Proposition 3.10. In particular vv ∗ v = v. (2) ⇒ (3): taking adjoints of both sides of the equality vv ∗ v = v we arrive at v ∗ vv ∗ = v ∗ . (3) ⇒ (4): multiplying both sides of v ∗ vv ∗ = v ∗ by v from the left we get vv ∗ vv ∗ = vv ∗ , i.e. the self-adjoint operator vv ∗ satisfies (vv ∗ )2 = vv ∗ . 

An operator v ∈ B(H) is called a partial isometry if it satisfies the equivalent conditions of Proposition 3.11. The projections v ∗ v and vv ∗ are referred to as the initial projection and final projection of the partial isometry v, while their ranges v ∗ v H and vv ∗ H are respectively the initial subspace and final subspace of v. Proposition 3.12 An operator v ∈ B(H) is a partial isometry if and only if there exists a closed subspace S ⊂ H such that ⎧ ⎨ ξ ,

vξ = ⎩0,

ξ ∈ S, ξ ∈ S ⊥.

(3.2)

In this case S is the initial subspace of v and the final subspace of v is v S . Proof Assume first that v is a partial isometry and let S = v ∗ v H. Then S is a closed subspace and for ξ ∈ S   vξ vξ  = ξ v ∗ vξ = ξ ξ  ,

3

27 3.4 · Polar Decomposition

while for ξ ∈ S ⊥   vξ vξ  = ξ v ∗ vξ = 0. In other words we obtain (3.2). Conversely, if v ∈ B(H) is an operator satisfying (3.2) for some closed subspace S ⊂ H then clearly v ∗ vξ = 0 for ξ ∈ S ⊥ . Moreover for ξ ∈ S we have v ∗ vξ = ξ . Indeed: take any η ∈ H and let η = ηS + ηS ⊥ be the decomposition of η into components in S and in S ⊥ . Then 

     η v ∗ vξ = ηS + ηS ⊥ v ∗ vξ = vηS + vηS ⊥ vξ

  = vηS vξ  = ηS ξ  = ηS + ηS ⊥ ξ = η ξ  .

In the third equality of the above computation we used the fact that v maps S ⊥ to {0} and in the fourth we used the property 

   vξ  vξ = ξ  ξ ,

ξ, ξ  ∈ S

which follows from the fact that v is isometric on S via the polarization identity.



A partial isometry v on H whose initial subspace is H (i.e. v ∗ v = 1) is called an isometry. This terminology is in agreement with the standard meaning of “isometry”, as in this case v is an isometric mapping of H onto the subspace v H = vv ∗ H of H.

3.4

Polar Decomposition

Theorem 3.13 Let x ∈ B(H). Then there exists a unique pair of operators (v, a) such that (1) x = va, (2) a is positive, (3) v ∗ v = s(a).

Proof 1 We begin by proving existence of a pair (v, a) satisfying conditions (1)–(3). Let a = (x ∗ x) 2 . For any ξ ∈ H we have     1 1

aξ 2 = (x ∗ x) 2 ξ (x ∗ x) 2 ξ = ξ x ∗ xξ = xξ xξ  = xξ 2 . It follows that there exists a linear map v00 : a H → x H such that for any ξ ∈ H v00 aξ = xξ.

28

3

Chapter 3 • Positive Operators

(indeed: if η ∈ a H, then there exists ξ ∈ H such that η = aξ and the vector xξ depends only on η because if η = aξ  for another ξ  ∈ H then xξ  −xξ = x(ξ −ξ  ) = a(ξ − ξ  ) =

η−η = 0). It is clear that v00 is an isometric map, and so it extends uniquely to an isometry v0 : a H → H. Now let v be the partial isometry defined as follows: ⎧ ⎨v ξ, 0 vξ = ⎩0,

ξ ∈ a H, ξ ∈ (a H)⊥ .

The operator v was defined precisely so that va = x. Moreover v ∗ v is the projection onto the initial subspace of v, i.e. onto the closure of the range of a. In other words v ∗ v = l(a) = s(a), because a is self-adjoint. We now pass to the proof of uniqueness of the pair (v, a). Let (u, b) be a pair of operators such that ▬ x = ub, ▬ b is positive, ▬ u∗ u = s(b). Then a 2 = x ∗ x = bu∗ ub = bs(b)b = b2 , so by uniqueness of square roots we get b = a. Now, since u is a partial isometry with initial projection s(b) = s(a), its initial subspace must be a H. Therefore uξ = 0,

ξ ∈ (a H)⊥ .

On the other hand, for η ∈ H we have u(aη) = xη = v0 (aη), so u coincides with v0 on a H. It follows that u = v.



The decomposition x = va of x ∈ B(H) obtained in Theorem 3.13 is called the polar decomposition of x. The partial isometry v entering polar decomposition is sometimes referred to as the phase of x, while the positive operator a is called the modulus or absolute value of x and is denoted by the symbol |x|. Let us further note that if x is self-adjoint then |x| = f (x), where f (λ) = |λ| for all λ ∈ σ (x).

29 3.5 · Monotone Convergence of Operators

ⓘ Remark 3.14 Let x ∈ B(H) and let x = v|x| be the polar decomposition of x. (1) The phase v of x is unitary if and only if ker x = 0 and the range of x is dense in H. (2) We have   x ∗ = |x|v ∗ = s |x| |x|v ∗ = v ∗ v|x|v ∗ = v ∗ v|x|v ∗ . Moreover, the operator v|x|v ∗ is positive and it is not hard to check that   s v|x|v ∗ = vs |x| v ∗ = vv ∗ vv ∗ = vv ∗ .  It follows that x ∗ = v ∗ v|x|v ∗ is the polar decomposition of x ∗ . (3) Assume that x is self-adjoint. Then |x| = x + + x − and v = s(x + ) − s(x − ).

3.5

Monotone Convergence of Operators

In addition to metric topology defined by the operator norm there are a number of other useful topologies on B(H). On of them is the strong topology defined by the family of seminorms {pξ }ξ ∈H , where x ∈ B(H).

pξ (x) = xξ ,

In this topology a net of operators (xi )i∈I converges to x ∈ B(H) if and only if for any ξ ∈ H we have xi ξ −→ xξ , i.e. the operators xi converge to x pointwise on H. i∈I

Theorem 3.15 Let (xi )i∈I be a net of self-adjoint operators such that

ij



⇒



xi ≥ xj



and assume that there exists a constant C > 0 such that xi ≤ C for all i ∈ I . Then the net (xi )i∈I has a supremum (least upper bound) in B(H), i.e. there exists a self-adjoint operator x such that x ≥ xi for all i ∈ I and any operator y such that y ≥ xi for all i ∈ I satisfies y ≥ x. Moreover, the net (xi )i∈I converges to x in strong topology.

Proof  We begin by noticing that for any ζ ∈ H the net ζ xi ζ  i∈I is bounded and nondecreasing, so it converges to its supremum sup ζ xi ζ  = lim ζ xi ζ  . i∈I

i∈I

3

30

Chapter 3 • Positive Operators

 It follows that for each ξ, η ∈ H the net ξ xi η i∈I converges, as

3

ξ xi η =

1 4

3 

  ik η + ik ξ xi (η + ik ξ ) ,

i ∈ I.

k=0

We let F (ξ, η) = lim ξ xi η ,

ξ, η ∈ H.

i∈I

It is easy to see that F is a sesquilinear form which is bounded, since       F (ξ, η) =  lim ξ xi η  = lim  ξ xi η  ≤ C ξ

η , i∈I

ξ, η ∈ H,

i∈I

and therefore there exists a unique operator x ∈ B(H) such that ξ xη = F (ξ, η) = lim ξ xi η ,

ξ, η ∈ H.

i∈I

The operator x is self-adjoint, since F (ξ, ξ ) is real for any ξ (as a limit of real numbers, cf. Proposition 3.3(2)). Moreover x = sup xi in the sense of partial order on B(H). Indeed: i∈I

for ξ ∈ H we have

    ξ xξ  = lim ξ xj ξ = sup ξ xj ξ ≥ ξ xi ξ  , j ∈I

j ∈I

i∈I

which means that x ≥ xi for all i ∈ I . Furthermore, if y ≥ xi for all i then for any ξ ∈ H we have ξ yξ  ≥ ξ xi ξ  ,

i ∈ I,

  so that ξ yξ  ≥ sup ξ xj ξ = ξ xξ . This means that y ≥ x. j ∈I

To see that (xi )i∈I converges to x in strong topology take any ξ ∈ H. We have 2 1 1 2

xξ − xi ξ 2 = (x − xi )ξ = (x − xi ) 2 (x − xi ) 2 ξ 1 2 1 2 ≤ (x − xi ) 2 (x − xi ) 2 ξ 1 2 = (x − xi ) (x − xi ) 2 ξ 1 2 ≤ 2C (x − xi ) 2 ξ   1 1 = 2C (x − xi ) 2 ξ (x − xi ) 2 ξ  = 2C ξ (x − xi )ξ  = 2C F (ξ, ξ ) − ξ xi ξ  −−→ 0, i∈I

so xi ξ −−→ xξ . i∈I



31 3.5 · Monotone Convergence of Operators

Notes In this chapter we have laid the groundwork for further development of operator theory. A particularly crucial role in several of the following chapters will be played by polar decomposition of operators. Numerous exercises and examples devoted to topics mentioned above can be found in [Ped, Section 3.2], [ReSi1 , Chapters VI and VII].

3

4

33

Spectral Theorems and Functional Calculus © Springer Nature Switzerland AG 2018 P. Sołtan, A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-319-92061-0_4

One of the most important facts about operators on finite dimensional spaces equipped with a scalar product is the spectral theorem which says that a self-adjoint operator m can be written as a linear combination m=

N 

(4.1)

λi Ei

i=1

of pairwise orthogonal projections {E1 , . . . , EN } whose coefficients {λ1 , . . . , λN } are the (real) eigenvalues of m. As a consequence, for any polynomial f ∈ C[ · ] we have f (m) =

N 

(4.2)

f (λi )Ei .

i=1

Indeed: since E1 , . . . , EN are pairwise orthogonal, for f (λ) = λn we obtain f (m) =

 N i=1

n λi Ei

=

N  i1 ,...,in =1

λi1 . . . λin Ei1 . . . Ein =

N 

λm i Ei

i=1

and the result for general f follows by linearity. No doubt, in case of infinite dimensional Hilbert spaces, the spectral theorem needs to be appropriately reformulated. The search for an analog of the decomposition (4.1) leads to the notion of a spectral measure which will be discussed in  Sect. 4.3. In this new formulation the linear combination (4.1) is replaced by an appropriately defined integral.

34

4

Chapter 4 • Spectral Theorems and Functional Calculus

It turns out that some of the very useful forms of the spectral theorem generalize not so much the equality (4.1), as (4.2). In other words the spectral theorem in one of its forms deals with functional calculus. Functional calculus for continuous functions on the spectrum of a self-adjoint operator was the focus of  Chap. 2. In this chapter we will broaden the class of functions for which functional calculus can be defined. As an intermediate step we will prove yet another formulation of the spectral theorem, namely we will show that a self-adjoint operator is unitarily equivalent to an operator of multiplication by a function on an L2 space. Each of the above mentioned results can be referred to as the spectral theorem and each has important and interesting consequences. At the end of the chapter we will define holomorphic functional calculus and give a brief account of functional calculus in C∗ -algebras.

4.1

Multiplication Operators

Let (, μ) be a semifinite measure space.1 Any f ∈ L∞ (, μ) defines a linear operator Mf on L2 (, μ): (Mf ψ)(ω) = f (ω)ψ(ω),

ω ∈ , ψ ∈ L2 (, μ).

The operator Mf is called the operator of multiplication by f (multiplication operator for short). Clearly Mf is bounded and Mf ≤ f ∞ . Lemma 4.1 For f ∈ L∞ (, μ) we have   ˙c .

f ∞ = inf c > 0 |f | ≤ Proof By definition

f ∞ = Let S =



inf

  sup f (ω).

μ( K )=0 ω∈

   ˙ c . We will show that inf C = sup |f (ω)| μ( K ) = 0 and C = c > 0 |f | ≤ ω∈

inf S. Let  s ∈ S. Then there exists a measurable s ⊂  such that μ( K s ) = 0 and s = sup f (ω). Thus |f | ≤ s on s which is of full measure and therefore s ∈ C. It follows ω∈s

that S ⊂ C, and so inf S ≥ inf C. Let c ∈ C. Then there is a measurable c ⊂  with μ( K c ) = 0 and |f | ≤ c on c .   Thus s = sup f (ω) ≤ c. In other words for any c ∈ C there exists s ∈ S such that s ≤ c, ω∈c

which shows that inf C ≥ inf S.



measure space (, μ) is semifinite if the measure μ is semifinite, i.e. for any measurable  ⊂  such that μ() > 0 there exists a measurable  ⊂  such that 0 < μ() + ∞. 1A

4

35 4.1 · Multiplication Operators

Let us note that L∞ (, μ) is a C∗ -algebra with usual operations of addition and multiplication and involution f → f . Proposition 4.2 The map  L∞ (, μ)  f −→ Mf ∈ B L2 (, μ) is an isometric ∗-isomorphism onto its range. Proof The fact that f → Mf preserves algebraic operations (including the involution) is rather obvious. We will check that the map in question is isometric. Let ψ ∈ L2 (, μ). For c ≥ ˙ |f |, so

f ∞ we have c ≥

f ψ 22 =



|f |2 |ψ|2 dμ ≤ c2 ψ 22 .



Therefore Mf ≤ c and consequently Mf ≤ f ∞ by Lemma 4.1. Conversely, if 0 < c < f ∞ then again by Lemma 4.1, the measure of the set   ω |f (ω)| ≥ c is non-zero. Thus there exists a measurable  ⊂  such that 0 < μ() < 1 +∞ and |f | ≥ c on . Let ψ = √μ() χ . Then ψ 2 = 1 and

f ψ 22 =

1 μ()



|f |2 dμ ≥ c2 ,



so that Mf ≥ c. It follows that Mf ≥ f ∞ .



For a measurable function f :  → C we define the essential range of f as    Vess (f ) = λ ∈ C for any neighborhood U of λ we have μ f −1 (U ) > 0 . Note that if f = ˙ g then Vess (f ) = Vess (g). Indeed: the condition that f = ˙ g means that f = g on certain measurable subset  ⊂  such that μ( K ) = 0. Therefore for any Borel subset A ⊂ C     μ f −1 (A) = μ f −1 (A) ∩  = μ ω ∈  f (ω) ∈ A     = μ ω ∈  g(ω) ∈ A = μ g −1 (A) ∩  = μ g −1 (A) . It is also easy to check that Vess (f ) is a closed subset of C. Theorem 4.3 Let f ∈ L∞ (, μ). Then σ (Mf ) = Vess (f ).

36

Chapter 4 • Spectral Theorems and Functional Calculus

Proof   Take a λ ∈ C KVess (f ). Then  there exists r > 0 such that the set ω |λ − f (ω)| < r has measure 0, i.e. λ − f (ω) ≥ r almost everywhere on . Therefore the function

4

g(ω) =

1 λ−f (ω) ,

ω∈

belongs to L∞ (, μ) and it is clear that g is the inverse of λ − f . Thus the operator Mg is the inverse of λ1 − Mf . Conversely, let λ ∈ Vess (f ). Then for any n the set   ω |λ − f (ω)| ≤ n1 has non-zero measure, and so there exists a measurable set n of finite and non-zero measure 1 such that |λ − f | ≤ n1 on n . Let ψn = √μ( χ . We have ψn 2 = 1 for all n and ) n n

     λ − f (ω) ψn (ω) ≤ 1 ψn (ω), n

ω ∈ ,

so that (λ1 − Mf )ψn 2 ≤ n1 . This, in turn, means that λ1 − Mf is not invertible, since if y were the inverse of λ1 − Mf then we would have 1 = y(λ1 − Mf )ψn 2 ≤ y (λ1 − Mf )ψn 2 −−−→ 0. n→∞



At some point later on the following fact will be needed: Proposition 4.4 Let (, μ) be a semifinite measure space and let f ∈ L∞ (, μ). Then  λ ∈ C is an eigenvalue of the operator Mf ∈ B L2 (, μ) if and only if there exists a measurable  ⊂  such that μ() > 0 and f = ˙ λ on . Proof Assume there exists a set  with the properties listed in the formulation of the theorem. Then it contains a measurable  such that 0 < μ( ) < +∞, and so ψ = √ 1  χ is an μ( )

element of L2 (, μ) such that ψ 2 = 1 and Mf ψ = λψ. On the other hand, if there exists a non-zero ψ ∈ L2 (, μ) such that Mf ψ = λψ then   fψ = ˙ λψ. Let  = ω ∈  ψ(ω) = 0 . Then we must have f = ˙ λ on . Moreover μ() > 0, since ψ = 0. 

As is easily checked any operator Mf on L2 (, μ) is normal and thus we immediately have the following corollary:   Corollary 4.5 We have Mf = sup |λ| λ ∈ Vess (f ) .

4

37 4.1 · Multiplication Operators

Proof  

Mf = |σ (Mf )| = sup |λ| λ ∈ σ (Mf ) .



ⓘ Remark 4.6 Assume that Mf is self-adjoint (i.e. f is real-valued). Let g be a continuous function on σ (Mf ) = Vess (f ). Then g(Mf ) = Mg◦f . Indeed: the mapping  C σ (Mf )  g −→ Mg◦f ∈ B(H) satisfies the conditions uniquely determining the continuous functional calculus for Mf (the fact that it is isometric follows from Corollary 4.5).

Theorem 4.7 Let x ∈ B(H) be a self-adjoint operator. Then there exists a semifinite measure space (, μ), an essentially bounded measurable real-valued function F on , and a unitary operator u : L2 (, μ) −→ H such that x = uMF u∗ .

Proof    Let A = f (x) f ∈ C σ (x) . For a vector ξ ∈ H we will denote by Aξ the subspace    f (x)ξ f ∈ C σ (x) ⊂ H. Let R be the family of orthonormal systems {ξi }i∈I in H such that Aξi ⊥ Aξj for any i, j ∈ I with i = j . We put a partial order on R via the relation of containment. By the Kuratowski-Zorn lemma there exists  a maximalelement {ξj }j ∈J of R.  Let us see that the subspace span Aξj is dense in H. Indeed: if there were a vector j ∈J  ξ with ξ = 1 and ξ ⊥ Aξj for all j then for any f, g ∈ C σ (x) 

     f (x)ξ g(x)ξj = ξ f (x)∗ g(x)ξj = ξ (f g)(x)ξj = 0,

j ∈ J,

and so the system {ξj }j ∈J ∪ {ξ } would also belong to R which would contradict the maximality of {ξj }j ∈J . Put  = J ×σ (x). By considering on J the discrete topology we obtain a locally compact topology on  (the product topology). Furthermore, let M be the family of subsets  ⊂ 

38

Chapter 4 • Spectral Theorems and Functional Calculus

such that for each j   j = λ ∈ σ (x) (j, λ) ∈ 

4

is a Borel subset of σ (x). Then M is a σ -algebra.2 For any j ∈ J the map    C σ (x)  f −→ ξj f (x)ξj ∈ C is a positive linear functional, so by the Riesz-Markov-Kakutani representation theorem there exists a positive regular Borel measure μj on σ (x) such that 

  ξj f (x)ξj = f dμj ,

 f ∈ C σ (x) .

σ (x)

The measures {μj }j ∈J allow us to define a measure μ on M: for  ∈ M we put μ() =



μj (j ).

j ∈J

It is easy to see that μ is a semifinite measure on  (all the measures μj are finite). Moreover ▬ ψ ∈ L2 (, μ) if and only if there exist j1 , j2 , . . . ∈ J such that j ∈ J K {j1 , j2 , . . . } we have ψ(j, λ) = 0 for μj -almost all λ and ∞     ψ(jn , λ)2 dμj (λ) < +∞, n n=1 σ (x)

▬ the space Cc (), i.e. the space of continuous functions on  with compact support, is dense in ∈ L2 (, μ).  We will now define the operator u ∈ B L2 (, μ), H . On the dense subspace Cc () we put uf =



f (j, x)ξj ,

f ∈ Cc ()

j ∈J

fact that M is a σ -algebra is rather obvious. Moreover, it turns out that M is nothing else, but the σ algebra of Borel subsets of . Indeed: clearly any open subset of  belongs to M, so that M contains all Borel sets. On the other hand if  is a Borel subset of  then for each j ∈ J the set j can be identified with   ∩ {j } × σ (x) . This identification comes from the homeomorphism 2 The

σ (x)  λ −→ (j, λ) ∈ {j } × σ (x), which, of course, preserves the Borel structure. It follows that each j is a Borel subset of σ (x).

39 4.1 · Multiplication Operators

(the sum is finite, as the support of f is compact). Then for any f ∈ Cc () we have  2   

uf = f (j, x)ξj = f (j, x)ξj f (i, x)ξi 2

j ∈J

j ∈J

i∈J

    f (j, x)ξj f (i, x)ξi = f (j, x)ξj f (j, x)ξj = i,j ∈J

=



j ∈J



ξj f (j, x)∗ f (j, x)ξj =

j ∈J

=

 



ξj |f |2 (j, x)ξj



j ∈J

|f | (j, λ) dμj (λ) = 2

j ∈J σ (x)



|f |2 dμ = f 22 ,



and consequently u extends to an isometry L2 (, μ) → H.  Note that the range of u contains Aξj which is linearly dense in H. This implies that j ∈J

u is surjective and, hence, unitary. Define F :  → C by F (j, λ) = λ,

(j, λ) ∈ .

Then F is bounded and continuous (so, in particular, measurable). Moreover, for any f ∈ Cc () ⊂ L2 (, μ) we have u(MF f ) = u(Ff ) = =

 j ∈J

 (Ff )(j, x)ξj j ∈J

xf (j, x)ξj = x



f (j, x)ξj = xuf.

j ∈J

By continuity we obtain uMF ψ = xuψ for all ψ ∈ L2 (, μ), which means that uMF u∗ = x. 

Two operators x ∈ B(H) and y ∈ B(K) are unitarily equivalent if there exists a unitary u ∈ B(K, H) such that x = uyu∗. Theorem 4.7 says that any self-adjoint operator is unitarily equivalent to an operator of multiplication by a bounded function (even by one which is continuous on a locally compact topological space). It turns out that analogous statement holds also for any normal operator (Theorem 7.6). Of course, operators which are unitarily equivalent have identical spectra and share many other properties. Also it is sometimes easy to answer questions about a given self-adjoint operator using the fact that it is equivalent to a multiplication operator. ⓘ Remark 4.8 (1) Let A ⊂ B(H) be a ∗-algebra of operators. A vector ξ ∈ H is cyclic for A if Aξ is a dense subspace of H. Now if x ∈ B(H) is a self-adjoint operator such that A = C∗ (x, 1) has a cyclic vector in H then by the reasoning in the proof of Theorem 4.7 we

4

40

4

Chapter 4 • Spectral Theorems and Functional Calculus

can show that x is unitarily equivalent to an operator of multiplication by the identity function on σ (x). (2) Let x ∈ B(H) and y ∈ B(K) be self-adjoint and unitarily equivalent via u ∈ B(K, H), i.e. x = uyu∗ . Then, as we already mentioned, we have σ (x) = σ (y) and for any  f ∈ C σ (x) we have f (x) = uf (y)u∗ . This follows immediately from the uniqueness of continuous functional calculus.

4.2

Borel Functional Calculus

Let x ∈ B(H) be a self-adjoint operator. In this section we will generalize the continuous functional calculus for x to a wider class of functions, namely the bounded Borel functions on σ (x). We begin with a proposition on automatic continuity of certain ∗homomorphisms.3 Proposition 4.9 Let K be a Hilbert space and let B be a Banach ∗-algebra with unit. Consider a unital ∗-homomorphism  : B → B(K). Then  is a contraction. Proof Let b ∈ B and λ ∈ C be such that λ1 − b is invertible in B. Then λ1 − (b) is an invertible operator. In particular, putting b∗ b instead of b we obtain    σ (b∗ b) ⊂ λ ∈ C λ1 − b∗ b is not invertible in B Therefore    (b) 2 = (b)∗ (b) = (b∗ b) = σ (b∗ b)    ≤ sup |λ| λ1 − b∗ b is not invertible in B ≤ b∗ b = b 2 , where the last equality follows from the fact that if |λ| > b∗ b then λ1 − b∗ b is invertible with (λ1 − b∗ b)−1 =

∞ 

λ−n−1 (b∗ b)n

n=0

(the series converges in the Banach algebra B). This argument shows that every unital ∗homomorphism from a Banach ∗-algebra to B(K) is contractive.



Note that if X is a topological space then the set B (X) of bounded Borel functions on X is a Banach ∗-algebra (in fact, a C∗ -algebra) with natural operations of addition and multiplication, involution f → f and the uniform norm · ∞ . ∗-homomorphism is a linear, multiplicative and ∗-preserving map. A homomorphism  : A → B between unital algebras is called unital when (1A ) = 1B . 3A

41 4.2 · Borel Functional Calculus

The extension of functional calculus for self-adjoint operators from the class of continuous functions to bounded Borel functions will be based on Theorem 4.7 and the observation made in Remark 4.6. Theorem 4.10 (Borel Functional Calculus)  Let x ∈ B(H) be a self-adjoint operator and denote by B σ (x) the ∗-algebra of bounded Borel functions on σ (x). Then there exists a unique unital ∗-homomorphism  B σ (x) → B(H) denoted by  B σ (x)  f −→ f (x) ∈ B(H) such that ▬ if f (λ) = λ for all λ ∈ σ (x) then f (x) = x, ▬ if (fn )n∈N is a uniformly bounded sequence of Borel functions converging pointwise to f then fn (x) −−−→ f (x) in strong topology. n→∞  Moreover this homomorphism extends the isomorphism C σ (x) → C∗ (x, 1) given by the continuous functional calculus.

Proof The operator x is unitarily equivalent to an operator of multiplication by a Borel (in fact even continuous) function F on a Hilbert space L2 (, μ). Moreover the range of F is σ (x), so for any bounded Borel function f on σ (x) the composition f ◦ F is measurable (with respect to the Borel structure on ) and bounded. Define  f ∈ B σ (x) .

f (x) = uMf ◦F u∗ ,

It is clear that f → f (x) is a unital ∗-homomorphism and if f (λ) = λ for all λ ∈ σ (x) then f (x) = x. Incidentally, it is also clear that f → f (x) is a contraction, as f (x) =

uMf ◦F u∗ = Mf ◦F ≤ f ∞ , so in this case we do not need to use Proposition 4.9. Let (fn )n∈N be a bounded sequence of Borel functions on σ (x) converging pointwise to f and take ξ ∈ H. Setting ψ = u∗ ξ and using the fact that u is isometric we obtain fn (x)ξ − f (x)ξ 2 = uM(f −f )◦F u∗ ξ 2 n 2 = uM(fn −f )◦F ψ 2 = M(fn −f )◦F ψ   2  2  = fn F (ω) − f F (ω)  ψ(ω) dμ(ω) −−−→ 0 

by the dominated convergence theorem.

n→∞

4

42

Chapter 4 • Spectral Theorems and Functional Calculus

 We have already established that the map B σ (x)  f → f (x) ∈ B(H) is a contractive unital ∗-homomorphism and that it maps the identity function to x. It is therefore immediate from the uniqueness of continuous functional calculus that it extends the latter. Let us address the uniqueness of the homomorphism

4

 B σ (x)  f −→ f (x) ∈ B(H).  Let  : B σ (x) → B(H) be a unital ∗-homomorphism (which by Proposition 4.9 is necessarily contractive) mapping the identity function to x and such that if (fn )n∈N is a bounded sequence of Borel functions converging pointwise to f then (fn ) converges to (f ) in strong topology. It follows from the uniqueness of continuous functional calculus that if f is continuous then (f ) = f (x). For an open set  ⊂ σ (x) choose a bounded sequence of continuous functions (fn )n∈N converging pointwise to χ . Then for any ξ ∈ H we have χ (x)ξ = lim fn (x)ξ = lim (fn )ξ = (χ )ξ. n→∞

n→∞

Thus the family   L =  ∈ M (χ ) = χ (x) contains all open sets. This family is also a λ-system (see Appendix A.2): (1)  is open, so  ∈ L, (2) if  ∈ L then (χ ) = (χ − χ ) = 1 − (χ ) = 1 − χ (x) = χ (x) which means that  ∈ L, (3) if (n )n∈N is a sequence of pairwise disjoint elements of L and  =

∞ 

n then

n=1

χ =

∞ 

χn = lim

N →∞

n=1

N 

χn ,

n=1

 i.e. χ is a pointwise limit of the sequence

N

n=1

(χ )ξ = lim

N →∞

N  n=1

(χn )ξ = lim

N →∞

N 

 χn

N ∈N

and for any ξ ∈ H

χn (x)ξ = χ (x)ξ.

n=1

Therefore  ∈ L. Since the family of all open subsets of σ (x) is a π-system, by Dynkin’s theorem on π- and λ-systems (Theorem A.2), we have M ⊂ L. Now any bounded Borel function is a pointwise

4

43 4.3 · Spectral Measures

limit of a bounded sequence of simple functions, so the homomorphisms  and f → f (x)  agree on all of B σ (x) .



4.3

Spectral Measures

Let  be a set and let  M be a σ -algebra of subsets of . A spectral measure on  is a map E : M → Proj B(H) such that ▬ E(∅) = 0, E() = 1, ), ▬ for any 1 , 2 ∈ M we have E(1 ∩ 2 ) = E( 2 1 )E( ∞ ∞ 

▬ for pairwise disjoint 1 , 2 , . . . ∈ M we have E n = E(n ). n=1

n=1

The sum in the last condition is taken to mean the limit of finite sums in strong topology. Note that it follows from the first two conditions that projections corresponding to disjoint sets are orthogonal and hence their sum is a projection. Fundamental examples of spectral measures arise from self-adjoint operators: let x ∈ B(H) be self-adjoint, put  = σ (x) and let M be the σ -algebra of Borel subsets of σ (x). Define  Ex : M   −→ χ (x) ∈ Proj B(H) .

(4.3)

Then Ex is a spectral measure. We will see later on that this measure determines x uniquely. Let E be a spectral measure on . Then for any ξ ∈ H the formula  −→ ξ E()ξ  ,

∈M

defines a finite (positive) measure on . We will denote this measure by the symbol ξ Eξ , so that the integral of a function f with respect to this measure will be written as   f dξ | Eξ  or f (ω) dξ | E(ω)ξ . 



Similarly, for ξ, η ∈ H the mapping ξ Eη :  −→ ξ E()η ,

∈M

is a complex measure with finite total variation. Let us recall that the total variation of N  

ν(n ), where {1 , . . . , N } a measure ν is the supremum of all sums of the form n=1

is a finite partition of  into measurable sets. for ν = ξ Eη the total variation is bounded by ξ

η . Indeed: for any partition {1 , . . . , N } there are complex numbers

44

Chapter 4 • Spectral Theorems and Functional Calculus

λ1 , . . . , λN of modulus one such that N N N         ξ Eη (n ) =  ξ E(n )η  = λn ξ E(n )η = ξ tη , n=1

4

n=1

where t =

N

n=1

λn E(n ). Since the projections E(1 ), . . . , E(N ) are pairwise

n=1

orthogonal, we have      N N λ E( ) λ E( )

t 2 = t ∗ t = n n m m n=1

m=1

N N   λn λm E(n )E(m ) = E(n ) = = 1 = 1 m,n=1

n=1

which shows that the total variation of ξ Eη is not greater than ξ

η . Let us notice further that for any bounded measurable function f on  the quantity



f (ω) dξ | E(ω)η



is a sesquilinear form with respect to the variables (ξ, η) which is bounded by

ξ

η

f ∞ . Consequently it defines a bounded operator xf ∈ B(H) via 

  ξ xf η = f (ω) dξ | E(ω)η,

ξ, η ∈ H.



Theorem 4.11 The map f → xf ∈ B(H) is a ∗-homomorphism from the algebra of bounded measurable functions on  into B(H). Moreover, if (fn )n∈N is a uniformly bounded sequence of functions converging pointwise to f then xfn −−−→ xf n→∞

in strong topology.

4

45 4.3 · Spectral Measures

Proof First let us notice that the map f → xf is a contraction for the norm · ∞ on bounded measurable functions and the operator norm, since

xf =

sup

ξ = η =1

   ξ xf η  =

sup

ξ = η =1

     f dξ | Eη  







≤ sup |f | · total variation of ξ Eη ≤ sup |f |. 



Therefore it will be enough to prove multiplicativity xf xg = xf g for simple functions only, as they are uniformly dense in the set of bounded measurable functions. Let f =

N 

αn χn ,

g=

n=1

M 

βm χm .

m=1

We have  xf xg =

N 

 αn E(n )

n=1

=



M 

 βm E(m )

m=1

αn βm E(n )E(m )

m,n

=



αn βm E(n ∩ m ) = xf g .

m,n

Analogous argument shows that xf = xf ∗ . Now let (fn )n∈N be a bounded sequence of measurable functions converging pointwise to f . Then for any ξ ∈ H 2    

xfn ξ − xf ξ 2 = ξ x|fn −f |2 ξ = fn (ω) − f (ω) dξ | E(ω)ξ  −−−→ 0 

by the dominated converge theorem.

n→∞



Let us introduce the following notation for operators xf : xf =

 f (ω) dE(ω).

(4.4)



Let x ∈ B(H) be self-adjoint and let Ex be the spectral measure on σ (x) defined by (4.3). Then it is easy to check that f → f (x) and f → xf agree on simple functions:

46

Chapter 4 • Spectral Theorems and Functional Calculus

for f =

N

λn χn both f (x) and xf are equal to

n=1 N 

4

λn Ex (n ).

n=1

Since any Borel function is a pointwise limit of simple functions, we get f (x) = xf for all f ∈ B σ (x) , i.e. f (x) =



 f ∈ B σ (x) .

f (λ) dEx (λ), σ (x)

In particular for f (λ) = λ we have x=

 λ dEx (λ).

(4.5)

σ (x)

For a fixed self-adjoint operator x ∈ B(H) a spectral measure Ex such that (4.5) holds is unique. To see that note that if x=

 λ dE(λ) σ (x)

for some Borel spectral measure E on σ (x) then the mapping

  B σ (x)  f −→ f (λ) dE(λ) σ (x)

satisfies the conditions uniquely defining the Borel functional calculus for x. Therefore for each bounded Borel function f we have





f (λ) dE(λ) = f (x) =

σ (x)

f (λ) dEx (λ). σ (x)

Applying this to f = χ for an arbitrary Borel set  ⊂ σ (x) we obtain E() = Ex (), so the two measures are equal.

47 4.4 · Holomorphic Functional Calculus

4.4

Holomorphic Functional Calculus

Let x ∈ B(H) be an arbitrary operator. In particular, x is not assumed to be self-adjoint. The algebraof functions holomorphic on a neighborhood of σ (x) will be denoted by the symbol H σ (x) .4 Let f ∈ H σ (x) and let  be a positively oriented curve surrounding σ (x) contained in the domain of holomorphy of f . Define f (x) =

1 2π i



f (λ)(λ1 − x)−1 dλ



(the integral is of a continuous Banach space-valued function over a compact set). As usual, the value of the integral is independent of the choice of the curve . Moreover, it is easy to see that f (x) commutes with x. Theorem 4.12 (Holomorphic Functional Calculus) Let x ∈ B(H). Then (1) if f is a polynomial function f (λ) = α0 + α1 λ + · · · + αn λn then f (x) = α0 1 + α1 x + · · · + αn x n ,  (2) the map H σ (x)  f → f (x) ∈ B(H) is a unital homomorphism.

Proof Let us first consider the case of f (λ) = λm for some m ∈ Z+ . Let  be a positively oriented circle around 0 with radius r > x . For λ ∈  we have (λ1 − x)−1 =

∞ 

λ−n−1 x n .

n=0

Therefore f (x) =

1 2π i



f (λ)(λ1 − x)−1 dλ



=

∞ 

1 2π i

n=0

=

∞   n=0



λm−n−1 x n dλ

 1 2π i



m−n−1

λ

 dλ x n = x m .



 σ (x) are equivalence classes of the equivalence relation identifying functions which coincide on some neighborhood of σ (x). 4 More precisely, elements of H

4

48

Chapter 4 • Spectral Theorems and Functional Calculus

As the map f → f (x) clearly is linear, we immediately infer that if f is a polynomial f (λ) = α0 + α1 λ + · · · + αn λn then f (x) = α0 1 + α1 x + · · · + αn x n .

4

 Now we will show that the map f → f (x) is multiplicative. Let f, g ∈ H σ (x) and let  and   be curves around σ (x) contained within the intersection of domains of holomorphy of f and g, and such that   lies outside of . Using the resolvent identity (λ1 − x)−1 − (μ1 − x)−1 = (μ − λ)(λ1 − x)−1 (μ1 − x)−1 (see Remark 1.3) we compute: f (x)g(x) =



1 2 2π i



   f (λ)(λ1 − x)−1 dλ g(μ)(μ1 − x)−1 dμ 



= = =

  

1 2π i 1 2π i 1 2π i

2  

f (λ)g(μ)(λ1 − x)−1 (μ1 − x)−1 dμ dλ

 

2  

 

2   



f (λ)g(μ)  (λ1 − x)−1 μ−λ

f (λ)g(μ) −1 μ−λ (λ1 − x) dμ dλ

− =



1 2π i

2 



1 2 2π i

f (λ)(λ1 − x)−1 −



1 2 2π i



1 2π i



f (λ) 

=

1 2π i





1 2 2π i

f (λ)(λ1 − x)−1



=

 



=

− (μ1 − x)−1 dμ dλ



    g(μ) μ−λ



f (λ)g(μ) μ−λ (μ1

 dμ dλ

g(μ)(μ1 − x)



g(μ) μ−λ

− x)−1 dμ dλ

−1

 

 dμ dλ



 f (λ) μ−λ dλ



=0

dμ 



 1 2π i

 

g(μ) μ−λ

 dμ (λ1 − x)−1 dλ

f (λ)g(λ)(λ1 − x)−1 dλ = (fg)(x).



The indicated integral is equal to 0 because μ ∈   and  lies inside   , so we are integrating a holomorphic function over a curve contractible within its domain of holomorphy. 

4

49 4.4 · Holomorphic Functional Calculus

  Proposition 4.13 Let x ∈ B(H). Then for any f ∈ H σ (x) we have σ f (x) =  f σ (x) . Proof   −1 Take μ ∈ C K f σ (x) . Then the function hμ : λ → μ − f (λ) is holomorphic on a neighborhood of σ (x) and we have   hμ (x) μ1 − f (x) = μ1 − f (x) hμ (x) = 1,    so that μ ∈ σ f (x) . In other words σ f (x) ⊂ f σ (x) .  Now let μ ∈ f σ (x) . Then there exists λ0 ∈ σ (x) such that μ = f (λ0 ) and (λ) μ − f (λ) = f (λ0 ) − f (λ) = (λ0 − λ) f (λλ00)−f = (λ0 − λ)g(λ), −λ

(4.6)

where g(λ) =

⎧ ⎨ f (λ0 )−f (λ) ,

λ = λ0 ,

⎩f  (λ ), 0

λ = λ0

λ0 −λ

 belongs to H σ (x) . Applying both sides of (4.6) to x we obtain μ1 − f (x) = (λ0 1 − x)g(x). If the operator μ1 − f (x) were invertible, so would have to be λ0 1 − x, as λ0 1 − x and g(x)   commute. However, λ0 ∈ σ (x), so it follows that μ ∈ σ f (x) . This shows that f σ (x) ⊂  σ f (x) . 

   Proposition 4.14 Let x ∈ B(H), g ∈ H σ (x) and f ∈ H σ g(x) . Then  f g(x) = (f ◦ g)(x) Proof Let  be a positively oriented curve surrounding σ (x) within domain of holomorphy of g and let   be a positively oriented curve in the domain of holomorphy of f surrounding   σ g(x) = g σ (x) and lying outside the image of  under g. For μ ∈   let hμ (λ) =  −1 1 . μ−g(λ) . Then hμ is holomorphic on a neighborhood of σ (x) and hμ (x) = μ1 − g(x) Thus  f g(x) = =

1 2π i 1 2π i



 −1 f (μ) μ1 − g(x) dμ





f (μ)hμ (x) dμ 

50

Chapter 4 • Spectral Theorems and Functional Calculus

= =

1 2π i

1 2π i



 



=

1 2π i





1 μ−g(λ) (λ1

 1 2π i



4

1 2π i

f (μ)  

f (μ) μ−g(λ)

 − x)−1 dλ dμ

 dμ (λ1 − x)−1 dλ

(f ◦ g)(λ)(λ1 − x)−1 dλ = (f ◦ g)(x).





Let D ⊂ C be an open set and denote by D the set {z z ∈ D}. Let f be a holomorphic function on D. Then we can define a function f : D → C putting f(z) = f (z),

z ∈ D.

It is easy to check that fis holomorphic on D.  Proposition 4.15 Let x ∈ B(H) and f ∈ H σ (x) . Then f (x)∗ = f(x ∗ ). Proof Choose a positively oriented curve  surrounding σ (x) within domain of holomorphy of f .  Denote by  the image of  under complex conjugation with negative orientation, and by  the same curve with positive orientation we obtain 



f (x) = =

1 2π i −1 2π i





f (λ)(λ1 − x)

=

−1 2π i

f (λ)(λ1 − x ∗ )−1 dλ

1 2π i

 − 

∗ dλ





=

−1

 

 f (μ)(μ1 − x ∗ )−1 dμ



f(μ)(μ1 − x ∗ )−1 dμ.





Proposition 4.16 Let x ∈ B(H) be self-adjoint. Then the holomorphic functional calculus  for x is the restriction to H σ (x) of the isomorphism  C σ (x) −→ C∗ (x, 1) given by the continuous functional calculus.

51 4.4 · Holomorphic Functional Calculus

Proof  For a temporary distinction, denote the image of f ∈ H σ (x) under holomorphic functional calculus by (f ) instead of f (x), reserving the latter symbol for the image of  f under continuous functional calculus. For f ∈ H σ (x) the operator (f ) is normal (by Proposition 4.15), and so      (f ) = σ (f )  = sup |f (λ)| λ ∈ σ (x) .   It follows that H σ (x)  f → (f ) extends uniquely to an isometry C σ (x) → B(H) mapping any polynomial p to p(x). But such a map must coincide with the continuous  functional calculus and hence both calculi must agree on H σ (x) . 

Consider now the entire function exp. For any x ∈ B(H) the operator exp(x) can be written as the sum of the convergent series exp(x) =

∞ 

1 m x m!

m=0

(this follows from the definition of holomorphic functional calculus and expansion of exp in a power series). By standard manipulation performed on absolutely convergent series we easily check that if x and y commute then:  exp(x) exp(y) =

∞ 



n=0

=

∞ 

1 k!

k=0

∞ 

xn n!

 ym m!

m=0 k   l=0

=

k ∞  

x l y k−l l! (k−l)!

k=0 l=0





k l k−l  (x+y)k xy = = exp(x + y). k! l k=0

The general situation is described by the next theorem:

Theorem 4.17 (Lie-Trotter Formula) Let x, y ∈ B(H). Then exp(x + y) = lim

n→∞



exp

1  1 n . n x exp n y

Proof For n ∈ N denote sn = exp

1

n (x

+ y) ,

tn = exp

1 1 n x exp n y .

4

52

Chapter 4 • Spectral Theorems and Functional Calculus

We have  ∞

sn =

1 1 m! nm (x

 ∞ + y)m ≤

m=0

4



1 1 m! nm

m

x + y = exp x + y

n

m=0

and





 

y

x + y

exp = exp .

tn ≤ exp n1 x exp n1 y ≤ exp x

n n n Furthermore  ∞

sn − tn =

1 1 m! nm (x

+ y)m −

m=0

 ∞

1 1 k k! nk x

  ∞

k=0

∞  1 = 1 + n (x + y) +

1 1 l l! nl y



l=0

+ y)m

1 1 m! nm (x

m=2

− 1 − n1 x − n1 y −

 ∞

1 1 k k! nk x

k=1

 ∞ =

1 1 m! nm (x

+ y)m −

m=2

∞  = n12

 ∞

1 1 m! nm−2 (x

+ y)m −

1 1 l l! nl y

1 1 l l! nl y



l=1

 ∞

k 1 1 k! nk−1 x

k=1

  ∞

l 1 1 l! nl−1 y

l=1

const. . n2

Using this and the identity snn − tnn =

n−1 

snr (sn − tn )tnn−1−r

r=0

we obtain the following estimate:

snn − tnn ≤

n−1 



l=1

  ∞

k=1

m=2

=

1 1 k k! nk x

  ∞

snr

sn − tn

tnn−1−r

r=0

  n ≤ n max sn , tn sn − tn ≤

const. n

 exp x + y



4

53 4.5 · Theorems of Fuglede and Putnam

and hence lim

n→∞



exp

1  1 n = lim tnn = lim snn n x exp n y n→∞

= lim

n→∞



n→∞

exp

1

n (x

+ y)

n

= exp(x + y). 

4.5

Theorems of Fuglede and Putnam

Theorem 4.18 (Fuglede’s Theorem) Let x and y be commuting elements of B(H). Assume further that x is normal. Then yx ∗ = x ∗ y.

Proof For λ ∈ C and let  u(λ) = exp λx ∗ − λx . Then u(λ) is a unitary operator, since we can easily check that u(λ)∗ = u(−λ) for all λ and u(λ)u(μ) = u(λ + μ),

λ, μ ∈ C.

Moreover u(0) = 1 and   u(λ) = exp(λx ∗ ) exp − λx = exp − λx exp(λx ∗ ) (as x is normal), so since x commutes with y, we see that   exp(−λx ∗ )y exp(λx ∗ ) = exp(−λx ∗ ) exp λx y exp − λx exp(λx ∗ ) = u(−λ)yu(λ). Therefore the holomorphic function h : C  λ −→ exp(−λx ∗ )y exp(λx ∗ ) ∈ B(H) is bounded by y . By Liouville’s theorem it is constant: exp(−λx ∗ )y exp(λx ∗ ) = h(0) = y,

λ∈C

54

Chapter 4 • Spectral Theorems and Functional Calculus

which can be rewritten as y exp(λx ∗ ) = exp(λx ∗ )y, i.e.

4

∞ 

λn ∗ n n! y(x )

n=0

=

∞ 

λn ∗ n n! (x ) y.

n=0

Equality of convergent power series guarantees equality of all coefficients, so in particular yx ∗ = x ∗ y. 

Note that if x ∈ B(H) is not normal then the conclusion of Theorem 4.18 fails already for y = x. Corollary 4.19 Let x1 , x2 , y ∈ B(H). Assume that x1 and x2 are normal and that yx1 = x2 y. Then yx1 ∗ = x2 ∗ y. Proof Identifying operators on H ⊕ H with 2 × 2 matrices over B(H) consider operators  x and  y on H ⊕ H given by   x=

 x1 0 0 x2

and  y=

00 . y0

x ∗ y which means that Then  x is normal and we have  y x = x y . By Fuglede’s theorem  y x∗ =  ∗ ∗ yx1 = x2 y. 

We say that two operators x1 , x2 ∈ B(H) are similar if there exists an invertible y ∈ B(H) such that yx1 y −1 = x2 . Corollary 4.19 allows us to prove that similar normal operators are, in fact, unitarily equivalent. Corollary 4.20 (Putnam’s Theorem) Let x1 , x2 ∈ B(H) be normal. If x1 and x2 are similar then they are unitarily equivalent. Proof There exists an invertible y ∈ B(H) such that yx1 y −1 = x2 , i.e. yx1 = x2 y. By the previous corollary we have yx1 ∗ = x2 ∗ y which can be rewritten as x1 y ∗ = y ∗ x2 . From this we infer that y ∗ yx1 = y ∗ x2 y = x1 y ∗ y,

4

55 4.6 · Functional Calculus in C*-algebras

1

and therefore also |y| = (y ∗ y) 2 commutes with x1 (cf. Remark 2.7). Let y = u|y| be the polar decomposition of y. Since both y and |y| are invertible, u is an invertible partial isometry. Hence u is unitary and, moreover, we have ux1 u∗ = u|y||y|−1 x1 |y||y|−1 u−1 = yx1 y −1 = x2 . 

Putnam’s theorem can be slightly strengthened in the following way: assume x1 , x2 ∈ B(H) are normal and y ∈ B(H) is such that yx1 = x2 y. Then, precisely as above we show that |y|x1 = x1 |y|. Thus x2 u|y| = x2 y = yx1 = u|x|x1 = ux1 |y|.

(4.7)

Now if ker y = {0} then also ker |y| = {0} and |y|H = (ker y)⊥ = H, so it follows from (4.7) that x2 u = ux1 . If we additionally assume that the range of y is dense in H then u is unitary (Remark 3.14(1)) and so it implements unitary equivalence of x1 and x2 . In other words we have: Corollary 4.21 Let x1 , x2 ∈ B(H) be normal. If there exists y ∈ B(H) such that yx1 = x2 y, ker y = {0} and y H = H then x1 and x2 are unitarily equivalent.

4.6

Functional Calculus in C*-algebras

Let A be a C∗ -algebra with unit. An element a ∈ A is called self-adjoint if a = a ∗ , while we say that a is normal if aa ∗ = a ∗ a. Continuing the analogy with B(H) we say that u ∈ A is unitary if u∗ u = uu∗ = 1. Notice that if u ∈ A is unitary and λ ∈ σ (u) (spectrum of a C∗ -algebra element was defined in  Sect. 1.3) then |λ| = 1. Indeed: first of all we have λ−1 ∈ σ (u−1 ), since if λ−1 1 − u−1 were invertible then so would be λ1 − u, as  (λ1 − u) − λ−1 u−1 (λ−1 1 − u−1 )−1  = − λ−1 u−1 (λ−1 1 − u−1 )−1 (λ1 − u) = 1.     Furthermore, since λ−1 ∈ σ (u∗ ) = σ (u) ⊂ z ∈ C |z| ≤ 1 and λ ∈ z ∈ C |z| ≤ 1 , we find that |λ| = 1. For any element a ∈ A and a function f holomorphic on a neighborhood of σ (a) we put f (a) =

1 2π i

 

f (λ)(λ1 − a)−1 dλ,

56

4

Chapter 4 • Spectral Theorems and Functional Calculus

where  is a positively oriented curve surrounding σ (a) contained within the domain of holomorphy of f . This way we define a unital homomorphism from the algebra of functions holomorphic on a neighborhood of σ (a) into A which maps polynomials onto  corresponding polynomials in a. Moreover, for a and f as above we have σ f (a) =  f σ (a) . Proofs of these facts are identical to those for A = B(H). Proposition 4.22 Let a be an element of a unital C∗ -algebra A. Then (1) if a is normal then |σ (a)| = a , (2) if a is self-adjoint then σ (a) ⊂ R. Proof The proof of (1) is established in exactly the same way as that of Proposition 2.3(1). Now let a be self-adjoint. Then we easily check that the element u = exp(ia) is unitary. By holomorphic functional calculus for a we see that λ ∈ σ (a) implies exp(iλ) ∈ σ (u), so that    exp(iλ) = 1, and hence λ ∈ R. 

Similarly as for holomorphic functions, we can also generalize the continuous functional calculus for operators to self-adjoint elements of C∗ -algebras:

Theorem 4.23 Let a ∈ A be self-adjoint. Then there exists a unique map  C σ (a)  f −→ f (a) ∈ A such that ▬ if f is a polynomial function f (λ) = α0 + α1 λ + · · · + αn λn then f (a) = α0 1 + α1 a + · · · + αn a n ,  ▬ f (a) = f ∞ for all f ∈ C σ (a) .  Moreover, this map is a ∗-isomorphism of the C∗ -algebra C σ (a) onto the smallest closed ∗-subalgebra of A containing 1 and a.

The proof of Theorem 4.23 is fully analogous to that of existence and uniqueness of continuous functional calculus for self-adjoint operators. We also have the following: Proposition 4.24 Let A be a unital C∗ -algebra and let a ∈ A be self-adjoint. Then  (1) if b is a self-adjoint element of A commuting with a then for any functions f ∈ C σ (a)  and g ∈ C σ (b) we have f (a)g(b) = g(b)f (a),  (2) if c ∈ A commutes with a then for any f ∈ C σ (a) we have f (a)c = cf (a),    (3) for any f ∈ C σ (a) we have σ f (a) = f σ (a) ,  (4) for any real-valued g ∈ C σ (a) the element g(a) ∈ A is self-adjoint and for any    f ∈ C g σ (a) we have f g(a) = (f ◦ g)(a).

57 4.6 · Functional Calculus in C*-algebras

As before, proofs of the facts listed above amount to repeating the arguments used for self-adjoint operators on a Hilbert space. An element a ∈ A is called positive if a is self-adjoint and σ (a) is contained in [0, +∞[. Using functional calculus we can easily show that any positive element has a unique positive square root. The notion of a positive element introduces a partial order on A: by definition a ≤ b if and only if b − a is positive. Proposition 4.25 Let a, b ∈ A be elements such that 0 ≤ a ≤ b. Assume that a is invertible. Then b is invertible and b−1 ≤ a −1 .

Once more the proof of this proposition is identical to that given for operators (Proposition 3.7).

Notes Further information on multiplication operators and Borel functional calculus can be found in the textbooks [Arv2 , ReSi1 , Rud2 ] and in [Hal, Section 7]. A more general view of holomorphic functional calculus from  Sect. 4.6 is presented in the monograph [Zel] and in [Arv1 ]. There, the reader will also find various examples and problems related to this topic. Theorem 4.7 as well as all results of  Sects. 4.2 and 4.3 for self-adjoint operators can be extended to apply to all normal operators. As we already mentioned in  Chap. 2, these extensions will be presented in  Sect. 7.4. Furthermore, functional calculus in C∗ algebras also extends to all normal elements, so that Theorem 4.23 and Proposition 4.24 hold for normal elements without substantial changes. However, the proofs of these results are best accomplished with techniques different from the ones used above for self-adjoint elements.

4

59

Compact Operators © Springer Nature Switzerland AG 2018 P. Sołtan, A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-319-92061-0_5

In this chapter we will discuss the most fundamental results on compact operators on Hilbert spaces. It is important to stress that analogous theory for operators on Banach spaces is somewhat more complicated and, in particular, requires more considerations of topological nature. The very definition of a compact operator between Banach spaces is significantly different from the one we will propose in the case of operators on a Hilbert space. In the more general framework of Banach spaces the definition of a compact operator amounts to the conclusion of Proposition 5.1. An interested reader will find more information on compact operators in any functional analysis textbook, e.g. in [Rud2 , Chapter 4].

5.1

Compact Operators on a Hilbert Space

We say that an operator x ∈ B(H) is finite dimensional if its range is finite a dimensional subspace of H. A compact operator on H is an element x ∈ B(H) which is a norm limit of finite dimensional operators. The sets of finite dimensional and compact operators will be denoted by F (H) and B0 (H) respectively. It is easy to check that both F (H) and B0 (H) are ideals in B(H), i.e. they are vector subspaces and the implications

x ∈ F (H), y ∈ B(H)



⇒



xy, yx ∈ F (H)



and

hold.

x ∈ B0 (H), y ∈ B(H)



⇒



xy, yx ∈ B0 (H)



5

60

Chapter 5 • Compact Operators

Any finite dimensional operator x is of the form xξ =

N 

αn (ξ )ψn ,

ξ ∈H

n=1

5

for some ψ1 , . . . , ψN ∈ H (basis of the range of x) and continuous linear functionals α1 , . . . , αN on H. It follows that αi (ξ ) = ϕi ξ  for some ϕ1 , . . . , ϕn ∈ H, so that x=

N 

ψn  ϕn .

n=1

Thus x∗ =

N 

ϕn  ψn

n=1

is also finite dimensional, and hence the ideal F (H) is closed under the involution of B(H). Consequently the ideal B0 (H) is closed under taking adjoints, since if x = lim xn and xn ∈ F (H) for all n then x ∗ = lim xn ∗ ∈ B0 (H). The ideal n→∞

n→∞

B0 (H) is moreover norm-closed. In particular if x ∈ B0 (H) is self-adjoint and f is a continuous function on σ (x) which can be uniformly approximated by polynomials without constant term then f (x) ∈ B0 (H). Proposition 5.1 An operator x ∈ B(H) is compact if and only if for any bounded set S ⊂ H the set x(S) is pre-compact.1 Proof Let us denote the closed unit ball of H by the symbol H1 . Clearly it is enough to prove that x is compact if and only if x(H1 ) is a pre-compact set. Let x be a compact operator and fix ε > 0. There exists a sequence (xn )n∈N of finite dimensional operators such that xn −−−→ x, and so there exists a finite dimensional operator n→∞

xn such that x − xn < 2ε . The set xn (H1 ) is pre-compact (it is bounded and contained in a finite dimensional subspace), so it has a finite ε2 -net {η1 , . . . , ηN }. Now for ξ ∈ H1 we have

xξ − ηk ≤ xξ − xn ξ + xn ξ − ηk ,

k ∈ {1, . . . , N}.

The term xξ − xn ξ is bounded by 2ε , and since xn ξ belongs to xn (H1 ), there exists k0 ∈ {1, . . . , N} such that xn ξ − ηk0 < 2ε . Thus xξ − ηk0 < ε, and so {η1 , . . . , ηN } is an ε-net for x(H1 ).

1 A subset

of a topological space is pre-compact if its closure is compact. A subset S of a complete metric space is pre-compact if and only if for any ε > 0 there exists a finite family of balls with radius ε covering S. The collection of centers of such balls is called an ε-net for S.

5

61 5.1 · Compact Operators on a Hilbert Space

ε } Now suppose that x ∈ B(H) is such that for any ε > 0 there is a finite ε-net {η1ε , . . . , ηN for x(H1 ). Let pε be the projection onto the finite dimensional subspace ε } span{η1ε , . . . , ηN

and let xε = pε x. Then xε ∈ F (H) and for any ξ ∈ H1 there exists k such that

xξ − ηkε < ε. Therefore

xξ − xε ξ = (1 − pε )xξ ≤ (1 − pε )(xξ − ηkε ) + (1 − pε )ηkε = (1 − pε )(xξ − ηkε ) ≤ 1 − pε

xξ − ηkε < ε, as the vector ηkε is orthogonal to the range of the projection 1 − pε . This way we have shown that for any ε there exists a finite dimensional operator within distance less than ε of x. It follows easily from this that x is a norm limit of finite dimensional operators. 

Let x ∈ B(H) be compact  and suppose that λ ∈ σ (x) is a non-zero eigenvalue. Then the eigenspace K = ξ ∈ H xξ = λξ for λ must be finite dimensional.  Indeed: the image under x of the bounded set K1 = ξ ∈ K ξ ≤ 1 is the ball   ξ ∈ K ξ ≤ |λ| which is not pre-compact unless dim K < +∞. This shows, in particular, that the operator 1 on an infinite dimensional Hilbert space is not compact. More generally a projection is compact if and only if it is finite dimensional. On the other hand the next proposition says that there are in a sense many compact operators. Proposition 5.2 For any Hilbert space H there exists a net of finite dimensional projections converging to 1 in strong topology. Proof Let {ψi }i∈I be an orthonormal basis of H and let J be the set of finite subsets of I directed by inclusion. For j ∈ J let pj =



ψi ψi .

i∈j

Then for any ξ ∈ H we have ξ =

ψi ξ  ψi and since the coefficients of ξ in the basis

i∈I

{ψi }i∈I are square-summable, we have

pj ξ − ξ 2 =

    ψi ξ  2 −−→ 0. j ∈J i∈I K j 

62

Chapter 5 • Compact Operators

5.2

Fredholm Alternative

Theorem 5.3 Let D be a connected open subset of C and let f : D → B(H) be a holomorphic function all of whose values are compact operators. Then either (a) for any z ∈ D the operator 1 − f (z) is not invertible or (b) the operator 1 − f (z) is invertible for z belonging to D K S, where S is a subset of D without an accumulation point in D; in this case for any z ∈ S the equation f (z)ψ = ψ has a non-zero solution.

5

Proof We note first that it is enough to prove that either (a) or (b) holds on a neighborhood of any given z0 ∈ D. Fix z0 ∈ D and let r > 0 be such that   Dr = z ∈ C |z − z0 | < r 1 is contained in D and z ∈ Dr implies 1 that f (z) − f (z0 ) < 2 . Let y be a finite dimensional operator such that f (z0 ) − y < 2 . Then for any z ∈ Dr we have f (z) − y < 1, and so 1 − f (z) + y is invertible and the function  −1 Dr  z −→ 1 − f (z) + y is holomorphic. The operator y can be written as y=

N 

ψn ϕn

(5.1)

n=1

for some vectors ϕ1 , . . . , ϕN , ψ1 , . . . ψN ∈ H and we can assume that the system {ψ1 , . . . , ψN } is linearly independent. For z ∈ Dr let ϕn (z) =

 −1 ∗ 1 − f (z) + y ϕn

and  −1 g(z) = y 1 − f (z) + y .

5

63 5.2 · Fredholm Alternative

Using (5.1) we find that g(z) =

N 

 −1 ψn  ϕn 1 − f (z) + y

n=1

=

N 

ψn

  −1 ∗ ϕn 1 − f (z) + y

(5.2)

n=1

=

N 

ψn  ϕn (z) .

n=1

It follows that for any z ∈ Dr we have g(z)H ⊂ y H. It is clear that   1 − g(z) 1 − f (z) + y = 1 − f (z) + y − y = 1 − f (z), and since the operator 1 − f (z) + y is invertible for z ∈ Dr , we see that ▬ 1 − f (z) is invertible if and only if 1 − g(z) is invertible,  ▬ the equation ψ = f (z)ψ has a non-zero solution (i.e. ker 1 − f (z) = {0}) if and only  if the equation ϕ = g(z)ϕ has a non-zero solution (ker 1 − g(z) = {0}). Suppose that ϕ = g(z)ϕ. Then ϕ belongs to the range of g(z) which is contained in the N

range of y, so that ϕ = βm ψm . In view of (5.2) we get m=1

βn =

N 

ϕn (z) ψm  βm .

(5.3)

m=1

Conversely, if β1 , . . . , βN satisfy (5.3) then ϕ =

N

βm ψm is a solution to the equation

m=1

ϕ = g(z)ϕ. It follows that this equation has a non-zero solution if and only if ⎛⎡





ϕ1 (z) ψ1  ⎥ ⎢ ⎜⎢ . .. ⎥ ⎢ ⎜ ⎢ d(z) = det ⎝⎣ . . ⎦ − ⎣ . ϕN (z) ψ1  1 1

⎤⎞ . . . ϕ1 (z) ψN  ⎥⎟ .. .. ⎥⎟ = 0. . . ⎦⎠ . . . ϕN (z) ψN 

  The function d is holomorphic on Dr , and consequently the set Sr = z ∈ Dr d(z) = 0 either has no accumulation points in Dr or Sr = Dr . Let us now investigate invertibility of 1 − g(z). For this we note that the equation  1 − g(z) ϕ = ξ

(5.4)

64

Chapter 5 • Compact Operators

has a solution for any ξ ∈ H if and only if the equation ϕ − g(z)ϕ = ζ has a solution for any ζ in the range of g(z). Indeed: substituting ϕ = ξ + ϕ

(5.5)

in (5.4) we obtain ξ + ϕ  − g(z)ξ − g(z)ϕ  = ξ,

5

hence ϕ  − g(z)ϕ  = g(z)ξ.

(5.6)

If ϕ  is a solution to this equation then ϕ = ξ + ϕ  is a solution of (5.4), and conversely, if ϕ satisfies (5.4) then ϕ  = ϕ − ξ satisfies (5.6). Moreover, as ξ runs over all of H, the vector ζ = g(z)ξ runs over the range of g(z). In other words, we can solve (5.4) for any ξ if and only if d(z) = 0. Summing up, for z ∈ Dr d(z) = 0 ⇐⇒



there exists ϕ = 0 such that ϕ = g(z)ϕ



and d(z) = 0 ⇐⇒



 for any ξ ∈ H there exists ϕ such that 1 − g(z) ϕ = ξ .

Thus 1 − g(z) is either not invertible on all of Dr (when S = Dr ) or Sr is a discrete subset of Dr and 1 − g(z) is a bijection H → H for z ∈ Dr K Sr . In the latter case, for −1 z ∈ Dr K Sr we have a formula for (1 − g(z) , namely (1 − g(z)

−1

  −1 ξ = 1 + 1 − g(z) y H g(z) ξ

 −1 (cf. (5.5)) which shows that 1 − g(z) is bounded.



Corollary 5.4 (Fredholm Alternative) Let x ∈ B0 (H). Then either 1 − x is invertible or the equation ψ = xψ has a non-zero solution. Proof Put D = C and f (z) = zx. Then the statement follows from Theorem 5.3 for z = 1.



Corollary 5.5 (Riesz-Schauder Theorem) Let x ∈ B0 (H). Then the set σ (x) does not have accumulation points other than possibly λ = 0. Moreover any non-zero element of σ (x) is an eigenvalue of finite multiplicity.

65 5.2 · Fredholm Alternative

Proof As in the proof of Corollary 5.4 we apply Theorem 5.3 to the function f (x) = zx on C. It follows that the set   S = z ∈ C the equation zxψ = ψ has a non-zero solution is discrete (it is not equal to C, as 0 ∈ S). Moreover, for z outside of S the operator 1 − zx is invertible, i.e. the operator  λ1 − x = λ 1 − λ−1 x is invertible provided λ = 0 and λ lies outside the discrete set   S −1 = z−1 z ∈ S ⊂ C K {0}, and any λ ∈ S −1 is an eigenvalue of x. The fact that non-zero eigenvalues of a compact operator must have finite multiplicity has already been discussed at the end of  Sect. 5.1. 

Note that since a discrete subset of C is necessarily countable, spectrum of a compact operator is a countable set. Corollary 5.6 (Hilbert-Schmidt Theorem) Let x ∈ B(H) be a compact normal operator. Then there exists an orthonormal basis of H consisting of eigenvectors of x. Proof Write the spectrum of x as σ (x) = {0} ∪ {λ1 , λ2 , . . . }, where for any n ∈ N we have λn = 0. Let {ψj0 }j ∈J be an orthonormal basis of ker x and for n ∈ N let {ψkn }k=1,...,dim ker(λn 1−x) be an orthonormal basis of the (finite dimensional) eigenspace of x for the eigenvalue λn . The union {ϕi }i∈I of all these bases is an orthonormal system in H. Define the subspace S = span{ϕi i ∈ I }. Both x and x ∗ leave S invariant, and hence they also leave S ⊥ invariant (cf. Proposition 2.1). Furthermore the operator x S ⊥ is compact, and consequently non-zero elements of its spectrum must be eigenvalues. But all eigenvectors  of x by definition belong to S ,   so σ x S ⊥ = {0}. As x S ⊥ is also normal, we have x S ⊥ = 0 because its spectral radius is equal to 0. This, however, shows that S ⊥ consists solely of eigenvectors of x (for the eigenvalue 0), so S ⊥ ⊂ S . It follows that S ⊥ = {0} or, in other words, S = H and we find that {ϕi }i∈I is an orthonormal basis of H. 

5

66

Chapter 5 • Compact Operators

Suppose a compact operator x has infinite spectrum and let {λn }n∈N be the sequence of all non-zero eigenvalues of x. Then λn −−−→ 0. Indeed: for any r > 0 the set n→∞     n |λn | > r is finite because for any n we have |λn | ≤ x , so that  λn |λn | ≥ r is a sequence of elements of a compact set z ∈ C r ≤ |z| ≤ x which cannot be infinite, since it does not have accumulation points.

5

Corollary 5.7 (Canonical Form of a Compact Operator) Let x ∈ B(H) be a compact operator. Then there exists a finite or countably infinite set N and orthonormal systems {ψn }n∈N and {ϕn }n∈N in H as well as a sequence {λn }n∈N of strictly positive numbers such that x=



λn ϕn  ψn ,

n∈N

with the sum convergent in norm. Proof 1 Let x = v|x| be the polar decomposition of x. Then the operator |x| = (x ∗ x) 2 is compact and positive. Therefore there exists an orthonormal basis of H consisting of eigenvectors of |x|. Let {ψn }n∈N be the subsystem of this basis consisting of eigenvectors corresponding to  ⊥ non-zero eigenvalues. Then {ψn }n∈N is an orthonormal basis of ker |x| . We have |x| =



λn ψn  ψn

n∈N

and the sum is norm convergent because for a finite F ⊂ N we have 

 λn ψn  ψn = fF |x| ,

n∈F

 where fF is the continuous function on σ |x| = {λn }n∈N given by ⎧ ⎨λ λ ∈ F, fF (λ) = ⎩0 λ ∈ F  and fF converges uniformly to the identity function on σ |x| as F ranges over the directed set of finite subsets of N (cf. remarks preceding Corollary 5.7).  ⊥ The partial isometry v maps the subspace ker |x| isometrically onto x H. Therefore putting ϕn = vψn for all n ∈ N we obtain another orthonormal system {ϕn }n∈N . Clearly x=v

 n∈N

λn ψn ψn =



λn ϕn  ψn .

n∈N



67 5.2 · Fredholm Alternative

Non-zero eigenvalues of |x| appearing in the canonical form of a compact operator x are called the singular values of the operator x. Note that the orthonormal systems in the canonical form of x are far from being unique.

Notes The study of compact operators between Banach spaces constituted the foundation of functional analysis and its applications with first examples of such operators coming from the theory of integral and differential equations. Examples of applications of the theory of compact operators in these areas of mathematics can be found e.g. in [Mau, Chapter VII], [Ped, Section 3.3], [ReSi1 , Chapter VI], while interesting problems and other examples concerning compact operators are available in the problem book [Hal].

5

6

69

The Trace © Springer Nature Switzerland AG 2018 P. Sołtan, A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-319-92061-0_6

We will now discuss the concept of the trace of an operator. This notion has applications in quantum physics as well as in many problems of pure mathematics. One of them concerns a precise description of the dual space of B0 (H) which we will give in detail. Analysis of the trace as an “unbounded linear functional” on B(H) leads to far reaching generalizations in the theory of operator algebras (under the guise of theory of weights). Our presentation of this topic will be based on the same foundations which underlie those generalizations and can be treated as an introduction to more advanced techniques of the theory of operator algebras.

6.1

Definition of the Trace

Let H be a Hilbert space and let {ξj }j ∈J be an orthonormal basis of H. For a positive t ∈ B(H) define Tr(t) =



 ξj tξj ∈ [0, +∞].

j ∈J

Proposition 6.1 For any x ∈ B(H) we have Tr(x ∗ x) = Tr(xx ∗ ). Proof For each i ∈ J we have  j ∈J

     ξi x ∗ ξj ξj xξi = xξi ξj ξj xξi j ∈J



  ξj xξi ξj = xξi j ∈J



  = xξi xξi  = ξi x ∗ xξi ,

(6.1)

70

Chapter 6 • The Trace

and hence Tr(x ∗ x) =

     ξi x ∗ xξi = ξi x ∗ ξj ξj xξi . i∈J j ∈J

i∈J

Now, since 

6

       2 ξi x ∗ ξj ξj xξi = xξi ξj ξj xξi =  ξj xξi  ≥ 0,

we can change the order of summation and using the calculation (6.1) in the fourth step (this time applying it to the sum over i and the operator x ∗ ), we obtain    Tr(x ∗ x) = ξi x ∗ ξj ξj xξi i∈J j ∈J

=



  ξi x ∗ ξj ξj xξi

j ∈J i∈J

=



  ξj xξi ξi x ∗ ξj

j ∈J i∈J

  = ξj xx ∗ ξj = Tr(xx ∗ ). j ∈J



Corollary 6.2 Let u, t ∈ B(H) with u unitary and t positive. Then Tr(utu∗ ) = Tr(t).

Proof 1 Put x = ut 2 . Then 1

1

x ∗ x = t 2 u∗ ut 2 = t

and

1

1

xx ∗ = ut 2 t 2 u∗ = utu∗ .

By Proposition 6.1 we have Tr(utu∗ ) = Tr(xx ∗ ) = Tr(x ∗ x) = Tr(t). Corollary 6.3 Let {ηj }j ∈J be another orthonormal basis of H and define Tr (t) =



 ηj tηj ,

t ∈ B(H)+ .

j ∈J

Then Tr = Tr. Proof There exists a unitary u ∈ B(H) such that uηj = ξj for all j ∈ J . Therefore Tr (t) =

 j ∈J

   u∗ ξj tu∗ ξj = ξj (utu∗ )ξj = Tr(utu∗ ) = Tr(t) j ∈J



6

71 6.2 · Trace Class and Hilbert-Schmidt Operators

for any t ∈ B(H)+ .



Corollary 6.3 shows that the function  ξi tξi  ∈ [0, +∞] Tr : B(H)+  t −→ i∈I

does not depend on the choice of the orthonormal basis {ξi }i∈I of H. We call this function the trace. The following properties of the trace are immediate from its definition: ▬ for t ∈ B(H)+ and λ ∈ R+ we have Tr(λt) = λ Tr(t), ▬ if t, r ∈ B(H)+ and t ≥ r then Tr(t) ≥ Tr(r), ▬ for t, r ∈ B(H)+ we have Tr(t + r) = Tr(t) + Tr(r), with the last property following from standard facts about series with positive terms. Proposition 6.4 For t ∈ B(H)+ we have Tr(t) ≥ t . Proof 1 Put r = t 2 . We have t = r ∗ r = r 2 . Take ε > 0 and let ψ ∈ H be such that ψ = 1 and

rψ > r − ε. Then ψ tψ = rψ 2 > r 2 − 2ε r + ε2 , so that 1

ψ tψ > t − 2ε t 2 + ε2 . Now let us choose an orthonormal basis {ξj }j ∈J of H such that ψ = ξj0 for some j0 ∈ J . Then 1

Tr(t) ≥ ψ tψ > t − 2ε t 2 + ε2 . As ε > 0 is arbitrary, we obtain Tr(t) ≥ t .

6.2



Trace Class and Hilbert-Schmidt Operators

Define   B1 (H) = span t ∈ B(H)+ Tr(t) < +∞ ,   B2 (H) = t ∈ B(H) Tr(t ∗ t) < +∞ .

(6.2)

72

Chapter 6 • The Trace

Elements of the subspace B1 (H) ⊂ B(H) are called the trace class operators, while B2 (H) is the set of Hilbert-Schmidt operators. Before proceeding with analysis of the classes B1 (H) and B2 (H) let us note that for any a, b ∈ B(H) we have ab =

1 4

3 

ik (b + ik 1)∗ a(b + ik 1).

(6.3)

k=0

Indeed:

6

3 

ik (b + ik 1)∗ a(b + ik 1) = (b + 1)∗ a(b + 1)

k=0

+ i(b + i1)∗ a(b + i1) − (b − 1)∗ a(b − 1) − i(b − i1)∗ a(b − i1) = b ∗ ab + b ∗ a + ab + a + ib ∗ ab + ab + ib ∗ a + ia − b ∗ ab + ab − b ∗ a − a − ib ∗ ab − ib ∗ a + ab − ia = 4ab.

 Lemma 6.5 Let x ∈ B(H) be such that Tr |x|p < +∞ for some p > 0. Then x ∈ B0 (H). Proof Let {ψj }j ∈J be an orthonormal basis of H. For any ε > 0 there exists a finite subset Jε ⊂ J such that 

 ψj |x|p ψj < ε.

j ∈Jε

Let pε be the projection onto span{ψj j ∈ Jε }. Then by Proposition 6.4 we have p p p |x| 2 − |x| 2 pε 2 = |x| 2 (1 − pε ) 2 = (1 − pε )|x|p (1 − pε )  ≤ Tr (1 − pε )|x|p (1 − pε )   ψj |x|p ψj < ε. = j ∈Jε

6

73 6.2 · Trace Class and Hilbert-Schmidt Operators

p

p

The operator |x| 2 pε is finite dimensional, and consequently |x| 2 is compact. Now since 2  p  |x| = f |x| 2 , where f (λ) = λ p for all λ ∈ σ |x| , it follows that also |x| ∈ B0 (H) and finally, writing x in its polar decomposition, we obtain x = u|x| ∈ B0 (H). 

Application of Lemma 6.5 to the case p = 2 immediately yields the third inclusion of the next theorem: Theorem 6.6 We have F (H) ⊂ B1 (H) ⊂ B2 (H) ⊂ B0 (H)

(6.4)

and each of these subsets is a self-adjoint ideal in B(H). Moreover    B1 (H) = x ∈ B(H) Tr |x| < +∞ .

(6.5)

Proof We begin by noticing that all sets in (6.4) are closed under taking adjoints. In  Sect. 5.1 we indicated that F (H) and B0 (H) are ideals in B(H). We will now show that B1 (H) is a right ideal in B(H). Take a positive a ∈ B(H) such that Tr(a) < +∞ and let b ∈ B(H). From (6.3) we know that ab =

1 4

3 

ik vk ∗ avk ,

k=0

where vk = (b + ik 1). For each k we have   1 1 1 1 Tr(vk ∗ avk ) = Tr vk ∗ a 2 a 2 vk = Tr (a 2 vk )∗ (a 2 vk )  1 1 1 1 = Tr (a 2 vk )(a 2 vk )∗ = Tr(a 2 vk vk ∗ a 2 )  ≤ Tr vk vk ∗ a = vk 2 Tr(a) < +∞, and hence ab ∈ B1 (H). Since the space B1 (H) is spanned by a as above, it is a right ideal in B(H). As a self-adjoint subset of B(H) it must also be a left ideal.1 This immediately shows that B1 (H) contains the set    x ∈ B(H) Tr |x| < +∞ .

right ideal J in B(H) closed under taking adjoints is also a left ideal: if x ∈ J and y ∈ B(H) then yx = (x ∗ y ∗ )∗ ∈ J. 1 Any

74

Chapter 6 • The Trace

 Indeed: if x ∈ B(H) and Tr |x| < +∞ then clearly |x| ∈ B1 (H) and writing x as x = u|x| (polar decomposition) we find that x ∈ B1 (H), as B1 (H) is an ideal. Conversely, if t ∈ B1 (H) then also |t| = u∗ t belongs to B1 (H) (here, again, t = u|t| is the polar decomposition). This means that |t| can be written in the form |t| =

n 

αi di

i=1

6

for some α1 , . . . , αn ∈ C and positive operators d1 , . . . , dn with finite trace. Since, as is easily checked, |t| ≤

n 

|αi |di ,

i=1 n    

we have Tr |t| ≤ |αi | Tr(di ) < +∞, i.e. t ∈ x ∈ B(H) Tr |x| < +∞ . This way i=1

we proved (6.5). We will now deal with the set B2 (H). It follows from the identity (a + b)∗ (a + b) + (a − b)∗ (a − b) = 2(a ∗ a + b∗ b),

a, b ∈ B(H)

that (a + b)∗ (a + b) ≤ 2(a ∗ a + b∗ b), and thus B2 (H) is a vector subspace of B(H). B2 (H) is, moreover, a left ideal, since if t ∈ B2 (H) then for any s ∈ B(H) we have (st)∗ (st) = t ∗ s ∗ st ≤ s 2 t ∗ t,  so that Tr (st)∗ (st) ≤ s 2 Tr(t ∗ t) < +∞, i.e. st ∈ B2 (H). As B2 (H) is a self-adjoint subset of B(H), it is also a right ideal. Now let x ∈ F (H). Then |x| is a positive finite dimensional operator.2 Therefore |x| has finite trace and consequently |x| ∈ B1 (H). The latter set is an ideal, so x = u|x| ∈ B1 (H) and we have proved that F (H) ⊂ B1 (H).

To finish the proof we only need to show that B1 (H) ⊂ B2 (H). Let x ∈ B1 (H). We have 1 1 ∗  1 1 1 1 x ∗ x = |x|2 = |x| 2 |x| 2 |x| 2 |x| 2 ≤ x |x| 2 |x| 2 = x |x|.

 Thus Tr(x ∗ x) ≤ x Tr |x| < +∞, i.e. B1 (H) ⊂ B2 (H).

2

Let us write x in the form x =

N

ψi ϕi , with {ψ1 , . . . , ψN } orthonormal. Then x ∗ x =

i=1



N

i=1

ϕi ϕi

is a positive operator on span{ϕ1 , . . . , ϕN }. Let {ξ1 , . . . , ξM } be an orthonormal basis of span{ϕ1 , . . . , ϕN } M 

consisting of eigenvectors of x ∗ x. Then x ∗ x = λj ξj ξj and by uniqueness of square roots we have j=1

|x| =

M j=1

 λj ξj ξj .

75 6.2 · Trace Class and Hilbert-Schmidt Operators





ⓘ Remark 6.7 We have t ∈ B(H)+ Tr(t) < ∞ = B1 (H) ∩ B(H)+ . Indeed: it follows from the definition (6.2) that   t ∈ B(H)+ Tr(t) < ∞ ⊂ B1 (H) ∩ B(H)+ . On the other hand, if a ∈ B1 (H) ∩ B(H)+ then a = |a| and    a ∈ x ∈ B(H) Tr |x| < +∞ .   Therefore t ∈ B(H)+ Tr(t) < ∞ ⊃ B1 (H) ∩ B(H)+ . Proposition 6.8 The function B1 (H) ∩ B(H)+  x → Tr(x) ∈ R+ extends uniquely to a linear functional on B1 (H). Proof  Since B1 (H) = span B1 (H) ∩ B(H)+ , it is enough to prove that an extension of Tr to B1 (H) exists. We will show that the formula N  i=1

αi xi −→

N 

αi Tr(xi )

i=1

provides one. To see this it is enough to check that if and x1 , . . . , xN ∈ B(H)+ with finite trace), then

N

N

αi xi = 0 (for some α1 , . . . , αN ∈ C

i=1

αi Tr(xi ) = 0.

i=1

Indeed: in this case we have N 

Re αi xi = 0 and

i=1

N 

Im αi xi = 0.

i=1

Let {1, . . . , N} = A ∪ B with A consisting of those i for which Re αi ≥ 0 and B of the remaining elements of {1, . . . , N} (so that Re αi < 0 for i ∈ B). Then  i∈A

Re αi xi =

 (− Re αi )xi i∈B

are linear combinations of positive operators with positive coefficients. Thus from additivity of the trace we obtain  i∈A

Re αi Tr(xi ) =

 (− Re αi ) Tr(xi ), i∈B

i.e. N  i=1

Re αi Tr(xi ) = 0.

6

76

Chapter 6 • The Trace

Similarly we get

N

Im αi Tr(xi ) = 0.



i=1

Let x ∈ B1 (H). The value of the linear extension of Tr from B1 (H) ∩ B(H)+ to B1 (H) on x will also be called the trace of x and just like for positive operators, we will also denote it by Tr(x). We will now show that the value of Tr(x) can be calculated by the same formula as for positive operators. Take any x ∈ B1 (H) and let {ξi }i∈I be an orthonormal basis of H. Using the polar 1 1 decomposition of x write x = u|x| = u|x| 2 |x| 2 . Then      ξi xξi   =  |x| 21 u∗ ξi |x| 21 ξi  ≤ |x| 21 u∗ ξi |x| 21 ξi .

6

Now, since  1 2  |x| 2 ξi = Tr |x| < +∞ i∈I

and  1  |x| 2 u∗ ξi 2 = Tr u|x|u∗ < +∞ i∈I

(i.e. |x| 2 and |x| 2 u∗ belong to B2 (H)), we obtain 1

1

   1 1  ξi xξi   ≤

|x| 2 u∗ ξi

|x| 2 ξi

i∈I

i∈I





|x| 2 u∗ ξi 2 1

 12  

i∈I

It follows that the series can be expressed as x =

1

|x| 2 ξi 2

 12

< +∞.

i∈I

i∈I N

ξi xξi  is absolutely convergent. Furthermore, because x αk xk , where α1 , . . . , αN ∈ C and the operators x1 , . . . , xN

k=1

are positive with finite trace, we have 

ξi xξi  =

i∈I



 ξi

i∈I

=

N  i∈I k=1

N 

 αk xk ξi

k=1

αk ξi xk ξi  =

N  k=1

αk



ξi xk ξi  ,

i∈I

and the last expression is independent of the choice of the basis {ξi }i∈I . This way we have established the following:

6

77 6.2 · Trace Class and Hilbert-Schmidt Operators

Proposition 6.9 Let x ∈ B1 (H) and let {ξi }i∈I be an orthonormal basis of H. Then the series 

ξi xξi 

(6.6)

i∈I

is absolutely convergent and its sum is independent of the choice of the basis {ξi }i∈I .

Since (6.6) is a linear functional on B1 (H) which coincides with Tr(·) on B1 (H) ∩ B(H)+ , we see that Tr(x) =



ξi xξi 

i∈I

for any x ∈ B1 (H) and any orthonormal basis {ξi }i∈I of H. ⓘ Remark 6.10 It is worth remembering that if x ∈ B(H) and for some orthonormal basis {ξi }i∈I of H the series (6.6) converges (absolutely) it does not necessarily mean that x ∈ B1 (H). For example let H be infinite dimensional, choose an orthonormal basis {ξi }i∈I of H and take u to be the unitary operator such that uξi = ξπ(i) , where π is

ξi uξi  a permutation of the set I with finite number of fixed points. Then the series is absolutely convergent, but |u| = 1 has infinite trace.

i∈I

Consider now x, y ∈ B2 (H). In a way similar to the proof of formula (6.3), we can prove the following analog of the polarization identity: x ∗y =

1 4

3 

ik (y + ik x)∗ (y + ik x).

(6.7)

k=0

of B(H), we have y + ik x ∈ B2 (H), and consequently AsB2 (H) is a vector subspace Tr (y + ik x)∗ (y + ik x) < +∞ for each k. In particular x ∗ y belongs to B1 (H).

Theorem 6.11 The ideal B2 (H) is a Hilbert space with scalar product x yTr = Tr(x ∗ y),

x, y ∈ B2 (H).

(6.8)

78

Chapter 6 • The Trace

Proof We already know that for x, y ∈ B2 (H) we have x ∗ y ∈ B1 (H). It follows that formula (6.8) makes sense and defines a sesquilinear form (linear in y and anti-linear in x) on B2 (H). This form is hermitian, as Tr(y ∗ x) =

   ∗   ψi x ∗ yψi  = Tr(x ∗ y) ψi y ∗ xψi = x yψi ψi = i∈I

i∈I

i∈I

for any orthonormal basis {ψi }i∈I of H. Moreover, it is strictly positive, since

6

x xTr = Tr(x ∗ x) ≥ x ∗ x = x 2 .

(6.9)

Let · 2 denote the norm on B2 (H) associated with · ·Tr . It remains to prove that  B2 (H), · 2 is a complete space. First note that if (xn )n∈N is a Cauchy sequence in B2 (H) then the inequality (6.9) shows that (xn )n∈N is also a Cauchy sequence for the operator norm. Let x be the limit of (xn )n∈N in B(H). In order to estimate x − xn 2 let us fix an orthonormal basis {ψi }i∈I of H. Now xm −−−− → x in norm and therefore also in strong topology. Therefore, for a finite subset m→∞

I0 ⊂ I we have   (x − xn )ψi 2 = lim (xm − xn )ψi 2 m→∞

i∈I0

i∈I0

≤ lim sup m→∞

 (xm − xn )ψi 2 = lim sup xm − xn 2 . 2 m→∞

i∈I

It follows that

x − xn 22 =

sup

 (x − xn )ψi 2 ≤ lim sup xm − xn 2 . 2

I0 ⊂I i∈I 0 |I0 | 0 and ξ ∈ H of norm 1 satisfying yξ > y −ε. 1 Furthermore let φ = yξ

yξ . Now let us complete the orthonormal system {φ} to a basis {φj }j ∈J of H. Then, putting x = ξ  φ , we have x 1 = 1 and, moreover,

6

         ψy (x) =  Tr(yx) =  φ yxφ j j   j ∈J

    ≥  φ yxφ  =  φ yξ   =

1



yξ yξ  = yξ > y − ε

which means that ψy ≥ y .

6.3



Hilbert-Schmidt Operators on L2

Let (, μ) be a σ -finite measure space such that the space L2 (, μ) is separable and let k ∈ L2 ( × , μ ⊗ μ). By Fubini’s theorem for almost all ω1 ∈  the function k(ω1 , ·) is square-integrable and integrating with over the variable ω1 the integral of the square of the absolute value of this function we obtain      k(ω1 , ω2 )2 dμ(ω2 ) dμ(ω1 ) = k 2 . 2 



In particular, for any ψ ∈ L2 (, μ) the integral

 k(ω1 , ω2 )ψ(ω2 ) dμ(ω2 ) 

makes sense for almost all ω1 and the resulting function of ω1 satisfies 2      k(ω1 , ω2 )ψ(ω2 ) dμ(ω2 ) dμ(ω1)   





  k(ω1 , ω2 )2 dμ(ω2 ) ψ 2 dμ(ω1 ) = k 2 ψ 2 . 2 2 2  

6

85 6.3 · Hilbert-Schmidt Operators on L2

From this we infer that the formula  (tk ψ)(ω1) = k(ω1 , ω2 )ψ(ω2 ) dμ(ω2),

ω1 ∈ 



defines an element tk ψ of L2 (, μ) and the resulting map tk : L2 (, μ)  ψ −→ tk ψ ∈ L2 (, μ) is linear and bounded with

tk ≤ k 2 .

(6.12)

The operator tk is called an integral operator, while the function k is the integral kernel of the operator tk . Theorem 6.18 The range of the map  L2 ( × , μ ⊗ μ)  k −→ tk ∈ B L2 (, μ)  coincides with B2 L2 (, μ) and the resulting operator  L2 ( × , μ ⊗ μ) −→ B2 L2 (, μ) is unitary.

Proof Let {ϕi }i∈I be an orthonormal basis of L2 (, μ). Then the system {ϕi ⊗ ϕj }i,j ∈I is an orthonormal basis of L2 (, μ) ⊗ L2 (, μ) ∼ = L2 ( × , μ ⊗ μ) (see Appendix A.3). This gives us a Fourier expansion k=



αi,j ϕi ⊗ ϕj .

i,j ∈I

Let I be the family of finite subsets of I . For A ∈ I let kA =

 i,j ∈A

αi,j ϕi ⊗ ϕj

86

Chapter 6 • The Trace

and consider the operator tkA . For each ψ ∈ L2 (, μ) we have 

(tkA ψ)(ω1 ) =



αi,j

i,j ∈A

ϕj (ω2 )ψ(ω2 ) dμ(ω2 ) ϕi (ω1 ), 

which means that tkA =



αi,j | ϕi ϕj |.

i,j ∈A

6

  In particular tkA ∈ F L2 (, μ) ⊂ B2 L2 (, μ) . Furthermore, it follows from the estimate (6.12) that

tk − tkA = tk−kA ≤ k − kA 2 −−→ 0, A∈I

so that tk is a compact operator. In addition, for any s ∈ I tk ϕs = lim tkA ϕs = A∈I



αi,s ϕi .

i∈I

Thus

tk ϕs 22 =



|αi,s |2

i∈I

and Tr(tk ∗ tk ) =

 s∈I

tk ϕs 22 =



|αi,s |2 = k 22 .

i,s∈I

The above arguments show that k → tk is an isometry from L2 ( × , μ ⊗ μ) into   B2 L2 (, μ) . Moreover, its range contains a dense subset F L2 (, μ) , and therefore this map must be unitary. 

Notes Trace class and Hilbert-Schmidt operators are very useful objects both in pure mathematics and in theoretical physics, where the duality between B1 (H) and B(H) provides a description of so called mixed states of a quantum system. As for other applications, let us point out that Theorem 6.18 can be regarded as a criterion of compactness. More precisely, if t is a bounded operator on L2 (, μ) which can be written as an integral operator with square-integrable kernel then t belongs to the class of Hilbert-Schmidt operators and hence is compact. Examples and exercises related to the trace, trace class operators and Hilbert-Schmidt operators can be found in [Mau, Chapter VII], [Ped, Section 3.4], [ReSi1 , Chapter VI] and the problem book [Hal].

87

Functional Calculus for Families of Operators © Springer Nature Switzerland AG 2018 P. Sołtan, A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-319-92061-0_7

In this chapter we will extend functional calculus to families of commuting self-adjoint operators. As an application of this extension we will be able to introduce in  Sect. 7.4 functional calculus for normal operators. This will be the only part of the book in which we will require some results of the theory of Banach algebras, or more specifically, C∗ -algebras. These have been gathered in Appendix A.5.

7.1

Holomorphic Functional Calculus

Let a1 , . . . , an ∈ B(H) be pairwise commuting operators. For a function f holomorphic n Ś σ (ai ) in Cn we define on a neighborhood of i=1

f (a1 , . . . , an )   n  · · · f (λ1 , . . . , λn )(λ1 1 − a1 )−1 · · · (λn 1 − an )−1 dλ1 . . . dλn , = 2π1 i n

1

where 1 , . . . , n are positively oriented curves (in C) such that for each i the curve i n Ś surrounds σ (ai ) and the product i lies in the domain of holomorphy of f . Of course i=1

the value of the integral does not depend on the choice of the curves  n 1 , . .. , n . Ś Just as in the case of a single operator, we will denote by H σ (ai ) the algebra of functions holomorphic on a neighborhood of

n Ś i=1

i=1

σ (ai ).

7

88

Chapter 7 • Functional Calculus for Families of Operators

Theorem 7.1 The map H

ą n

 σ (ai )  f −→ f (a1 , . . . , an ) ∈ B(H)

(7.1)

i=1

is a unital homomorphism.

7

Proof It is clear that the map (7.1) is linear. Now we check that for any k ∈ {1, . . . , n} and N ∈ Z+ N if f (λ1 , . . . , λn ) = λN k then f (a1 , . . . , an ) = ak : choose 1 , . . . , n , so that for each i the curve i lies in   z ∈ C |z| > ai . We have 

1 n 2π i



···

n

=



−1 λN · · · (λn 1 − an )−1 dλ1 . . . dλn k (λ1 1 − a1 )

1



1 n 2π i



···

n

=

. j =k

=

λN k

 ∞



1 2π i

∞ 

   ∞ λ1−i−1 a1i · · · λn−i−1 ani dλ1 . . . dλn

i=0

1

i=0

1 2π i

i=0

   ∞   −i−1 i λj−i−1 aji dλj · 2π1 i λN a dλ k k j

j i=0

∞ . j =k



k i=0

  ∞  λj−i−1 dλj aji · i=0

j

1 2π i



−i−1 λN dλk aki j

 = akN .

k

It remains to prove multiplicativity of the map (7.1). First we notice that resolvents of commuting operators commute. To see this note that since a1 , . . . , an pairwise commute, for any λi ∈ ρ(ai ) the product of resolvents (λi 1 − ai )−1 in arbitrary order is the inverse of n / (λi 1 − ai ). The remaining calculation is similar to the one carried out in the proof of i=1  n  Ś σ (ai ) and let 1 , . . . , n and 1 , . . . , n be positively Theorem 4.12: let f, g ∈ H i=1

oriented curves surrounding the spectra of σ (a1 ), . . . , σ (an ) such that the sets n Ś i=1

n Ś

i and

i=1

i lie in the intersection of domains of holomorphy of f and g. Furthermore we ask that

for each i the curve i lie outside of i .

89 7.1 · Holomorphic Functional Calculus

Using the resolvent identity (1.2) we compute f (a1 , . . . , an )g(a1 , . . . , an ) =



1 n 2π i

 n

· =





1 n 2π i

1 2n 2π i



···



f (λ1 , . . . , λn )

i=1

1





···

n

g(μ1 , . . . , μn ) 1

 

···

n .

(μj 1 − aj )−1 dμ1 . . . dμn

j =1

···

1 n

n

n . (λi 1 − ai )−1 dλ1 . . . dλn



f (λ1 , . . . , λn )g(μ1 , . . . , μn ) 1

n .  (λi 1 − ai )−1 (μi 1 − ai )−1 dλ1 . . . dλn dμ1 . . . dμn i=1

=



1 2n 2π i n  .



 

···

···



f (λ1 , . . . , λn )g(μ1 , . . . , μn )

n

1 n

1

1 μi −λi

  (λi 1 − ai )−1 − (μi 1 − ai )−1 dλ1 . . . dλn dμ1 . . . dμn .

i=1

The quantity n  .

1 μi −λi



(λi 1 − ai )−1 − (μi 1 − ai )−1



i=1

is a linear combination of terms which are products of a number of factors (λi 1 − ai )−1 and a number of factors of the form (μj 1 − aj )−1 . For example if n = 3 we have 3  .

1 μi −λi



(λi 1 − ai )

−1

− (μi 1 − ai )

i=1

=

1 1 1 μ1 −λ1 μ2 −λ2 μ3 −λ3

−1



 (λ1 1 − a1 )−1 (λ2 1 − a2 )−1 (λ3 1 − a3 )−1

− (μ1 1 − a1 )−1 (λ2 1 − a2 )−1 (λ3 1 − a3 )−1 − (λ1 1 − a1 )−1 (μ2 1 − a2 )−1 (λ3 1 − a3 )−1 − (λ1 1 − a1 )−1 (λ2 1 − a2 )−1 (μ3 1 − a3 )−1 + (μ1 1 − a1 )−1 (μ2 1 − a2 )−1 (λ3 1 − a3 )−1 + (μ1 1 − a1 )−1 (λ2 1 − a2 )−1 (μ3 1 − a3 )−1 + (λ1 1 − a1 )−1 (μ2 1 − a2 )−1 (μ3 1 − a3 )−1

− (μ1 1 − a1 )−1 (μ2 1 − a2 )−1 (μ3 1 − a3 )−1 .

7

90

Chapter 7 • Functional Calculus for Families of Operators

The curves 1 , . . . , n and 1 , . . . , n are chosen so that the integral of any term containing at least one (μi 1 − ai )−1 is equal to 0. To see this, consider e.g. the term       3 2 1 3

2

1

1 f (λ1 , λ2 , λ3 )g(μ1 , μ2 , μ3 ) μ1 −λ 1

1 1 μ2 −λ2 μ3 −λ3

(μ1 1 − a1 )−1 (λ2 1 − a2 )−1 (λ3 1 − a3 )−1 dλ1 dλ2 dλ3 dμ1 dμ2 dμ3        f (λ1 ,λ2 ,λ3 ) 1 1 = μ1 −λ1 dλ1 μ2 −λ2 μ3 −λ3 g(μ1 , μ2 , μ3 ) 3 2 3 2 1

1

(μ1 1 − a1 )−1 (λ2 1 − a2 )−1 (λ3 1 − a3 )−1 dλ2 dλ3 dμ1 dμ2 dμ3 .

7

The integral

 1

λ1 −→

f (λ1 ,λ2 ,λ3 ) μ1 −λ1 dλ1

equals 0 because the function

f (λ1 ,λ2 ,λ3 ) μ1 −λ1

is holomorphic in a contractible region containing the curve 1 . It follows that we can eliminate from the expression we established for f (a1 , . . . , an ) g(a1 , . . . , an ) all terms containing at least one (μi 1 − ai )−1 . This way we obtain f (a1 , . . . , an )g(a1 , . . . , an )     2n  = 2π1 i ··· · · · f (λ1 , . . . , λn )g(μ1 , . . . , μn ) n

1 n

n . i=1

=



1 n 2π i



···

n

1

1 μi −λi

n .



f (λ1 , . . . , λn ) 1

(λj 1 − aj )−1 dλ1 . . . dλn dμ1 . . . dμn

j =1

 

1 n 2π i



···

n n .

 1

 g(μ1 ,...,μn ) dμ1 . . . dμn n / (μi −λi ) i=1

(λj 1 − aj )−1 dλ1 . . . dλn

j =1

=



1 n 2π i

 n

···



f (λ1 , . . . , λn )g(λ1 , . . . , λn ) 1

n .

(λj 1 − aj )−1 dλ1 . . . dλn

j =1

= (fg)(a1 , . . . , an ) by the multidimensional version of Cauchy’s formula.



The holomorphic functional calculus for finite families of commuting self-adjoint operators we defined above has several properties analogous to those of functional

7

91 7.2 · Continuous Functional Calculus

calculus for one operator. The next lemma describes a weak version of the spectral mapping theorem: Lemma 7.2 Let a1 , . . . , an ∈ B(H) be pairwise commuting operators and let P be a polynomial in n variables. Then ą  n  σ P (a1 , . . . , an ) ⊂ P σ (ai ) i=1

  = P (λ1 , . . . , λn ) λi ∈ σ (ai ), i = 1, . . . , n . 

Proof Take μ ∈ P

n Ś

 σ (ai ) . Then the function

i=1

 −1 f (z1 , . . . , zn ) = μ − P (z1 , . . . , zn ) is holomorphic on a neighborhood of

n Ś

σ (ai ) in Cn and we have

i=1

  μ1 − P (a1 , . . . , an ) f (a1 , . . . , an ) = f (a1 , . . . , an ) μ1 − P (a1 , . . . , an ) = 1.  Consequently μ ∈ σ P (a1 , . . . , an ) .



Lemma 7.2 will be of crucial importance in the next section where we define continuous functional calculus for families of commuting self-adjoint operators.

7.2

Continuous Functional Calculus

Theorem 7.3 Let x1 , . . . , xn ∈ B(H) be pairwise commuting self-adjoint operators. Then there exists a unique unital ∗-homomorphism ą  n C σ (ai ) −→ B(H), i=1

denoted by f → f (x1 , . . . , xn ), such that if prj :

n Ś i=1

onto j th coordinate then prj (x1 , . . . , xn ) = xj ,

j = 1, . . . , n.

σ (ai ) → σ (aj ) is the projection

92

Chapter 7 • Functional Calculus for Families of Operators

Proof If a ∗-homomorphism described in the theorem exists then it must assign the value  P (x1 , . . . , xn ) = αi1 ,...,in x1i1 · · · xnin

(7.2)

i1 ,...,in

to the polynomial P (λ1 , . . . , λn ) =



αi1 ,...,in λi11 · · · λinn .

i1 ,...,in

Let us, therefore, take (7.2) as the definition of P (x1 , . . . , xn ). It is easy to see that the right hand side of (7.2) is a normal operator, and so, by Lemma 7.2

7

   P (x1 , . . . , xn ) = σ P (x1 , . . . , xn )    ≤ sup |P (λ1 , . . . , λn )| λi ∈ σ (xi ), i = 1, . . . , n = P ∞ ,  where · ∞ denotes the uniform norm on C depends only on the values of P on the set

n Ś

n Ś

(7.3)

 σ (ai ) . In particular P (x1 , . . . , xn )

i=1

σ (ai ).

i=1

It is immediate that the map P −→ P (x1 , . . . , xn ) is a ∗-homomorphism the estimate (7.3) shows that it extends uniquely to a ∗ n and  Ś homomorphism C σ (ai ) → B(H) satisfying the conditions of the theorem.

 i=1

7.3

Joint Spectrum

Consider a family x1 , . . . , xn ∈ B(H) of pairwise commuting self-adjoint operators and n Ś the compact space X = σ (xi ). Let i=1

 J = f ∈ C(X) f (x1 , . . . , xn ) = 0 . 

Clearly J is an ideal in C(X). It is shown in Appendix A.5.2 that we can associate with J a closed subset Y ⊂ X such that J = C0 (X K Y )

and

C(X)/J ∼ = C(Y ).

The mapping C(X)  f → f (x1 , . . . , xn ) factorizes through C(X)/J (since J is, by definition, its kernel), and so we obtain an isometric unital ∗-homomorphism C(Y ) −→ B(H)

7

93 7.4 · Functional Calculus for Normal Operators

which we call the continuous functional calculus for the operators x1 , . . . , xn . The set Y is the joint spectrum of the operators x1 , . . . , xn which we denote by the symbol σ (x1 , . . . , xn ). n Ś Clearly, a point (μ1 , . . . , μn ) ∈ σ (xi ) lies outside of σ (x1 , . . . , xn ) if and only if  n i=1  Ś there exists a function f ∈ C σ (xi ) such that i=1

f (μ1 , . . . , μn ) = 0

7.4

and f (x1 , . . . , xn ) = 0.

Functional Calculus for Normal Operators

Let x ∈ B(H). Define Re x = 12 (x + x ∗ ) and Im x = are self-adjoint and

1 (x 2i

− x ∗ ). Then Re x and Im x

x = Re x + i Im x.

(7.4)

The operators Re x and Im x are usually called the real part and imaginary part of x. Moreover, (7.4) is clearly the unique way to write x as a sum of a self-adjoint operator and an anti-self-adjoint operator (i.e. a self-adjoint operator multiplied by i). It is easy to check that x is normal if and only if its real and imaginary parts commute. Proposition 7.4 Let x ∈ B(H) be a normal operator. Then   σ (x) = a + ib (a, b) ∈ σ (Re x, Im x) .

(7.5)

Proof Take (c, d) ∈ σ (Re x, Im x). Then the function f : (a, b) −→

1 (c+id)−(a+ib)

is continuous on σ (Re x, Im x) and it easily follows that the operator (c + id)1 − x = (c + id)1 − (Re x + i Im x) is invertible with inverse f (Re x, Im x). This proves the containment “⊂” in (7.5). Let λ ∈ ρ(x) and put r = Re λ, s = Im λ. Suppose further that (r, s) ∈ σ (Re x, Im x). For ε > 0 let f be a continuous function on σ (Re x, Im x) such that f ∞ ≤ 1, f (r, s) = 1 and the support of f is contained in the set   (p, g) (p − r)2 + (q − s)2 ≤ ε2 .

94

Chapter 7 • Functional Calculus for Families of Operators

We have   −1 f (Re x, Im x) = f (Re x, Im x) (r + is)1 − x (r + is)1 − x , and hence   f (Re x, Im x) ≤ f (Re x, Im x) (r + is)1 − x (r + is)1 − x −1 . Note that (r + is)1 − x = (r + is)1 − (Re x + i Im x) = g(Re x, Im x), where g(p, q) = (r + is) − (p + iq). Therefore

7

 f (Re x, Im x) (r + is)1 − x = f (Re x, Im x)g(Re x, Im x) = (fg)(Re x, Im x) ≤ fg ∞ ≤ ε (as |g| ≤ ε on the support of f ) and it follows that  f (Re x, Im x) ≤ ε (r + is)1 − x −1 .

(7.6)

But the right hand side (7.6) is arbitrarily small (as we vary ε), while f (Re x, Im x) ≥ 1, because f (r, s) = 1. This contradiction shows that if λ = r + is ∈ σ (x) then (r, s) ∈ σ (Re x, Im x) and we get the containment “⊃” in (7.5). 

Given a normal operator x ∈ B(H) and a continuous function f on σ (x) we can define f (x) = f˘(Re x, Im x), where f˘(u, v) = f (u + iv). Corollary 7.5 Let x ∈ B(H) be a normal operator. Then there exists a unique unital  ∗-homomorphism C σ (x) → B(H) denoted by  C σ (x)  f −→ f (x) ∈ B(H) such that if f (λ) = λ for all λ ∈ σ (x) then f (x) = x. Moreover f → f (x) is an isometric  ∗-isomorphism of C σ (x) onto C∗ (x, 1). Proof By uniqueness of the decomposition of x into its real and imaginary parts, the condition f (x) = x for the identity function f (λ) = λ is equivalent to the condition that f1 (x) = Re x

and

f2 (x) = Im x,

7

95 7.4 · Functional Calculus for Normal Operators

where f1 = Re f and f2 = Im f . The continuous functional calculus for the normal operator x provides an isometric ∗-homomorphism  C σ (Re x, Im x)  f −→ f (Re x, Im x) ∈ B(H)

(7.7)

satisfying this condition. Moreover, since polynomials in Re x and Im x span a dense   subalgebra of C σ (Re x, Im x) = C σ (x) , the condition determines the homomorphism uniquely. Finally note that the range of (7.7) contains all polynomials in x and x ∗ . These are dense in C∗ (x, 1) and the range of a ∗-homomorphism between C∗ -algebras is always closed by Theorem A.12, so it follows that the range of (7.7) is C∗ (x, 1). 

The ∗-isomorphism described in Corollary 7.5 is called the continuous functional calculus for the normal operator x. In the same way as for self-adjoint operators we prove the following properties of the continuous functional  calculus: ▬ the spectral mapping theorem: σ f (x) = f σ (x) for any normal x ∈ B(H) and  f ∈ C σ (x) ,   ▬ for any g ∈ C σ (x) the operator g(x) is normal and for f ∈ C σ g(x) we have  f g(x) = (f ◦ g)(x).  Using the isomorphism C σ (x)  f → f (x) ∈ C∗ (x, 1) we can immediately prove an analog of Theorem 4.7 for normal operators: Theorem 7.6 Let x ∈ B(H) be a normal operator. Then there exist a semifinite measure space (, μ), an essentially bounded measurable function F on  and a unitary operator  u ∈ B L2 (, μ), H such that x = uMF u∗ .

The proof of Theorem 7.6 is identical to the proof of the analogous Theorem 4.7. The key element is the possibility of applying functions continuous on σ (x) to x. Theorem 7.6 allows us to extend the Borel functional calculus to the class of normal operators: Theorem 7.7  Let x ∈ B(H) be a normal operator and denote by B σ (x) the algebra of bounded Borel functions on σ (x). Then there exists a unique unital ∗-homomorphism B → B(H) denoted by  B σ (x)  f −→ f (x) ∈ B(H)

(Continued )

96

Chapter 7 • Functional Calculus for Families of Operators

Theorem 7.7 (continued) such that ▬ if f (λ) = λ for all λ ∈ σ (x) then f (x) = x, ▬ if (fn )n∈N is a uniformly bounded sequence of Borel functions converging pointwise to f then fn (x) −−−→ f (x) in strong topology. n→∞  Moreover the above homomorphism extends the isomorphism C σ (x) → C∗ (x, 1) given by the continuous functional calculus.

7

Just as in the case of Theorem 7.6, the proof of Theorem 7.7 is merely a repetition of the steps taken to prove the analogous statement for self-adjoint operators (Theorem 4.10). Let x ∈ B(H) be a normal operator. The extension of functional calculus for normal operators to all bounded Borel functions on σ (x) makes it possible to define the spectral measure Ex associated to x by the formula Ex () = χ (x),

 ∈ M,

where M is the σ -algebra of Borel subsets of σ (x). It is not hard to show (again, repeating the proof for self-adjoint operators, cf.  Sect. 4.3) that for f ∈ B σ (x) we have  f (x) = f (λ) dEx (λ) σ (x)

and, in particular, x=

 λ dEx (λ).

(7.8)

σ (x)

Finally, as is the case of self-adjoint operators, the spectral measure Ex such that (7.8) holds is unique.

Notes Functional calculus and other versions of the spectral theorem for normal operators are usually introduced within the framework of Banach algebras and, in particular, Gelfand’s theory of commutative Banach algebras ([Arv2 , Chapters 1 and 2], [Mau, Chapter VIII], [Ped, Chapter 4], [Rud2 , Chapters 10 and 11], [Zel]). Particularly relevant is the theory of commutative C∗ -algebras with the famous theorem of Gelfand and Naimark which says that any commutative C∗ -algebra with unit is isometrically ∗isomorphic to an algebra of continuous functions on a uniquely determined compact space.

97 7.4 · Functional Calculus for Normal Operators

Our approach does not fully avoid Banach algebras, but we minimize their use favoring a more direct analysis of normal operators. Further developments include functional calculus for families of commuting normal operators which is developed by methods introduced in this chapter.

7

Part II

Unbounded Operators

101

Operators and Their Graphs © Springer Nature Switzerland AG 2018 P. Sołtan, A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-319-92061-0_8

Applications of operator theory in other branches of mathematics and in mathematical physics very often involve operators which are not bounded. This poses numerous difficulties whose source is, for the most part, the lack of useful algebraic structure on the set of unbounded operators. Our presentation of the theory of unbounded operators on a Hilbert space will focus on a few select issues and our preferred strategy for dealing with them will be to reduce them to questions about bounded operators. We will begin with some introductory information gathered in  Sect. 8.1. In the following chapters we will introduce our key tool which we call the z-transform and later use this tool to extend various versions of the spectral theorem to unbounded self-adjoint operators. The final chapters will be devoted to several classical topics like self-adjoint extensions of symmetric operators and elements of the theory of one-parameter groups of unitary operators.

8.1

Basics of Unbounded Operators

Let H be a Hilbert space. The space H ⊕ H has a natural Hilbert space structure with scalar product:  ξ η

  ψ = ξ ψ + η φ , φ

ξ, ψ, η, φ ∈ H.

In this part of the book we will look at operators on H from a slightly different perspective than we have done so far. From now on a linear operator T on H will be defined on a vector subspace D(T ) ⊂ H called the domain of the operator T . In other words T is a linear map T : D(T ) −→ H.

8

102

Chapter 8 • Operators and Their Graphs

We will not be assuming that D(T ) = H, but except for some very general considerations and the material of  Sect. 11.2, we will be assuming that D(T ) is dense in H. In the latter case we say that T is densely defined. The graph of the operator T is the subspace 1

0 G(T ) =

8

ψ Tψ

ψ ∈ D(T ) ⊂ H ⊕ H.

We call T closed if the subspace G(T ) is closed in H ⊕ H. Note that for any x ∈ B(H) we have D(x) = H and x is closed. Moreover, the closed graph theorem (Corollary A.7) says that if T is closed and D(T ) = H then T is bounded. The condition that an operator T be closed replaces to a certain extent the condition of continuity. More precisely, it is easy to see that T is closed if and only if for any sequence (ψn )n∈N of elements of D(T ) we have

lim ψn = ψ, lim T ψn = φ

n→∞

n→∞



⇒



ψ ∈ D(T ), T ψ = φ .

It is worthwhile to characterize subspaces which are graphs of operators: Proposition 8.1 A subspace G ⊂ H ⊕ H is a graph of a linear operator if and only if it does not contain non-zero vectors of the form  0 . η

(8.1)

Proof Clearly a graph of a linear map cannot contain such vectors. On the other hand, if G does not contain non-zero vectors of the form (8.1) then the condition 

 ξ ξ , ∈G η1 η2

implies η1 = η2 , because G is a vector subspace. Therefore G is the graph of a map T defined on 1 0  ξ ∈G D(T ) = ξ ∈ H there exists η ∈ H such that η by T ξ = η, where η is the unique vector such that is linear.

2ξ 3 η

Vectors of the form (8.1) will be called vertical.

∈ G. It is easy to check that this map 

8

103 8.1 · Basics of Unbounded Operators

Let T be an operator on H. We say that T is closable if G(T ) is a graph of an operator, i.e. when G(T ) does not contain non-zero vertical vectors. In this case the operator whose graph is G(T ) is called the closure of T and is denoted by T . Now if S and T are operators on H then S is an extension of T , or S contains T , if G(T ) ⊂ G(S). This means that D(T ) ⊂ D(S) and for ψ ∈ D(T ) we have Sψ = T ψ. In this case we write T ⊂ S. In particular, if T is closable then T is an extension of T : T ⊂ T. Proposition 8.2 An operator T is closable if and only if for any sequence (ψn )n∈N of elements of D(T ) we have

lim ψn = 0, lim T ψn = φ

n→∞

n→∞



⇒



φ=0 .

(8.2)

Proof If T is closable then condition (8.2) follows immediately from the fact that T is closed. On 2 3 the other hand, if G(T ) contains a non-zero vertical vector η0 ∈ G then there exists a sequence (ψn )n∈N of elements of D(T ) such that lim ψn = 0 and

n→∞

lim T ψn = η = 0.

n→∞



Proposition 8.3 Let T be an operator on H. Then T is densely defined if and only if G(T )⊥ does not contain any non-zero vectors of the form  ξ . 0

(8.3)

Proof 2 3 Suppose a non-zero vector ξ is orthogonal to D(T ). Then clearly the vector ξ0 is orthogonal to G(T ). Conversely, the condition  ξ ⊥ G(T ) 0 implies that ξ is orthogonal to D(T ), so if ξ = 0 then D(T ) is not dense in H.



Vectors of the form (8.3) will be called horizontal. Propositions 8.1 and 8.3 yield the following corollary: Corollary 8.4 A subspace G ⊂ H ⊕ H is the graph of a closed densely defined operator if and only if G is closed, does not contain non-zero vertical vectors and G⊥ does not contain non-zero horizontal vectors.

104

Chapter 8 • Operators and Their Graphs

Let T be a closed operator. The graph G(T ) of T is then a closed subspace of H ⊕ H, so it is itself a Hilbert space. Moreover, the map  D(T )  ψ −→

ψ ∈ G(T ) Tψ

is bijective. We can therefore transfer the Hilbert space structure of G(T ) onto D(T ) defining the scalar product by 

 ψ Tψ

ψ φT =



φ Tφ

ψ, φ ∈ D(T ).

,

The resulting norm on D(T ) is then given by

8

ψ T =

-

ψ 2 + T ψ 2 ,

ψ ∈ D(T )

and it is called the graph norm. With this norm D(T ) is a Hilbert space. ⓘ Remark 8.5 The graph norm can be defined for any linear operator—not necessarily closed one. In fact, it is easy to see that an operator T on H is closed if and only if D(T ) is complete in the graph norm.

As an application of Remark 8.5 we can consider a generalization of the notion of a multiplication operator introduced in  Sect. 4.1. Let (, μ) be a semifinite measure space and let f be a measurable function on . Define Mf to be the operator on L2 (, μ) such that   D(Mf ) = ψ ∈ L2 (, μ) f ψ ∈ L2 (, μ) and for ψ ∈ D(Mf ) we have Mf ψ = f ψ. Proposition 8.6 Mf is a closed operator. If f is finite almost everywhere then Mf is densely defined. Proof The graph norm on D(Mf ) is given by

ψ Mf =

 

|ψ| dμ + 2

 

1  1 2 2  2 2 |f | |ψ| dμ = 1 + |f | |ψ| dμ , 2

2



 i.e. it coincides with the norm of the Hilbert space L2 , (1 + |f |2 )μ . Clearly ψ ∈ D(Mf )  if and only if ψ ∈ L2 , (1 + |f |2 )μ , so D(Mf ) is a Hilbert space in its graph norm.   Now let χn be the characteristic function of ω ∈  |f (ω)| ≤ n . If f is finite almost everywhere then the sequence of functions (χn )n∈N converges pointwise to 1 almost

8

105 8.2 · Adjoint Operator

everywhere. Moreover the range of each operator Mχn is contained in D(Mf ) and by the dominated convergence theorem

φ − Mχn φ 2 −−−→ 0,

φ ∈ L2 (, μ).

n→∞



It follows that D(Mf ) is dense in L2 (, μ).

8.2

Adjoint Operator

Let T be a densely defined operator on H. Let G ⊂ H ⊕ H be a subspace defined as follows:   

ξ (8.4) ∈ G ⇐⇒ ∀ ψ ∈ D(T ) ξ T ψ = η ψ . η 2 3 Then G is a graph of an operator on H. Indeed: if η0 ∈ G then η is orthogonal to D(T ), and so η = 0. The operator whose graph is G defined by (8.4) is denoted by T ∗ and we call it the adjoint of T . Note that the description of continuous linear functionals on Hilbert spaces shows that the domain of T ∗ consists precisely of those vectors ξ for which the functional D(T )  ψ −→ ξ T ψ is bounded. We say that an operator T is self-adjoint if T = T ∗ . This equality means, in particular, that D(T ) = D(T ∗ ). When T ⊂ T ∗ then T is called symmetric or hermitian. Let us examine the graph of T ∗ . We have      ξ ξ ∈ G(T ∗ ) ⇐⇒ ∀ ψ ∈ D(T ) η η

 Tψ −ψ



 =0 .

In other words        ξ ξ 0 1 ∗ ∈ G(T ) ⇐⇒ ⊥ G(T ) , η η −1 0 i.e.  ∗

G(T ) =

0 1 G(T ) −1 0

⊥

 =

0 1  G(T )⊥ −1 0

(8.5)

106

Chapter 8 • Operators and Their Graphs

2 0 13 (the last equality follows from the fact that −1 0 is a unitary operator on H ⊕ H). Since the orthogonal complement of any subset is closed, it follows that T ∗ is always closed. In particular a self-adjoint operator is automatically closed. Proposition 8.7 A densely defined operator T is closable if and only if T ∗ is densely defined. Proof The operator T is closable if and only if G(T ) does not contain non-zero vertical vectors, 2 0 13 which is equivalent to −1 0 G(T ) not containing non-zero horizontal vectors. However, 

8

0 1 G(T ) = G(T ∗ )⊥ , −1 0

2 0 13 ∗ ⊥ so −1 0 G(T ) does not contain non-zero horizontal vectors if and only if G(T ) does not contain non-zero horizontal vectors which, by Proposition 8.3, is equivalent to the fact that D(T ∗ ) is dense. 

In particular, if T is closable then T ∗ is densely defined, so there exists the operator (T ∗ )∗ which we usually denote by T ∗∗ . We have 

   ⊥ 0 1  0 1 0 1  ∗ ⊥ G(T ) = = G(T ). G(T ) = G(T )⊥ −1 0 −1 0 −1 0 ∗∗

Therefore T ∗∗ = T . Formula (8.5) and elementary properties of the orthogonal complement give the following corollary: Corollary 8.8 Let S and T be densely defined operators on H such that T ⊂ S. Then S ∗ ⊂ T ∗.

It is important to note the crucial difference between the adjoint operator for bounded and unbounded operators. In the former case we can always write   ξ xη = x ∗ ξ η , while in the latter, the formula   φ T ψ = T ∗ φ ψ can be used only after making sure that φ ∈ D(T ∗ ).

107 8.3 · Algebraic Operations

ⓘ Remark 8.9 Let us bring attention to the following fact: a self-adjoint operator does not have proper symmetric extensions. Indeed: if T = T ∗ , T ⊂ S and S ⊂ S ∗ then S ⊂ S ∗ ⊂ T ∗ = T ⊂ S, so that S = T .

8.3

Algebraic Operations

Algebraic operations on unbounded operators can become rather technically involved. Let T and S be operators on H. Then the sum S + T is defined on the domain D(S + T ) = D(S) ∩ D(T ) and (S + T )ψ = Sψ + T ψ,

ψ ∈ D(S + T ).

Next we define the product (composition) ST setting   D(ST ) = ψ ∈ D(T ) T ψ ∈ D(S) and (ST )ψ = S(T ψ),

ψ ∈ D(ST ).

It turns out that the sum or the product of densely defined operators might not be densely defined (see [Kat, Chapter 6 §1.4] or [ReSi1 , Section VIII.1]). Also the sum or product of closed operators might fail to be closed (or even closable). Nevertheless, if x ∈ B(H) and T is densely defined that, of course, T + x is densely defined. Similarly, when u ∈ B(H) then uT is densely defined and so is T u if u is invertible. Proposition 8.10 Let T be a closed operator on H and let x ∈ B(H). Then the operator T + x is closed. Proof Let (ψn )n∈N be a sequence of elements of D(T + x) = D(T ) such that ψn −−−→ ψ n→∞

and

T ψn + xψn −−−→ φ n→∞

for some ψ, φ ∈ H. Clearly we then have T ψn −−−→ φ − xψ. As T is closed, we have n→∞

ψ ∈ D(T ) and T ψ = φ − xψ. But this means that ψ ∈ D(T + x) and (T + x)ψ = φ. In particular T + x is closed.

 Proposition 8.11 Let T be a closed operator on H and let u ∈ B(H). Then (1) T u is closed, (2) if u is invertible then uT is closed.

8

108

Chapter 8 • Operators and Their Graphs

Proof Ad (1). Take a sequence (ψn )n∈N of elements of D(T u) and assume that ψn −−−→ ψ n→∞

and

T uψn −−−→ φ. n→∞

Let ψn = uψn . Then for each n we have ψn ∈ D(T ), ψn −−−→ uψ and n→∞

T ψn −−−→ φ. n→∞

As T is closed, it follows that uψ ∈ D(T ) and T (uψ) = φ. In other words ψ ∈ D(T u) and (T u)ψ = φ which shows that T u is closed. Ad (2). If (ψn )n∈N is a sequence of elements of D(uT ) = D(T ) such that

8

ψn −−−→ ψ n→∞

and

uT ψn −−−→ φ, n→∞

then the sequence (T ψn )n∈N satisfies T ψn = u−1 uT ψn −−−→ u−1 φ. n→∞

Thus, by closedness of T , we have ψ ∈ D(T ) and T ψ = u−1 φ. In other words ψ ∈ D(uT ) and uT ψ = φ, which means that uT is closed. 

Corollary 8.12 Let T be a closed operator on H and let x ∈ B(H) be such that x H ⊂ D(T ). Then T x is bounded. Proof By Proposition 8.11(1) the operator T x is closed. Moreover D(T x) = H, so by the closed graph theorem T x is bounded. 

Proposition 8.13 Let S and T be densely defined operators on H such that ST is densely defined.1 Then T ∗ S ∗ ⊂ (ST )∗ . Proof Take ψ ∈ D(T ∗ S ∗ ). Then for any ξ ∈ D(ST ) we have     ψ ST ξ  = S ∗ ψ T ξ = T ∗ S ∗ ψ ξ , since ψ ∈ D(S ∗ ) and S ∗ ψ ∈ D(T ∗ ). The right hand side of the above equality is continuous  with respect to ξ , and so ψ ∈ D (ST )∗ and (ST )∗ ψ = T ∗ S ∗ ψ. 

1 Note

that it does not follow from density of D(ST ) that the domain of S is dense. Consider e.g. T = 0.

8

109 8.3 · Algebraic Operations

Proposition 8.14 Let T be a densely defined operator on H and let x ∈ B(H). Then (xT )∗ = T ∗ x ∗ . Proof  We already know that T ∗ x ∗ ⊂ (xT )∗ . Take η ∈ D (xT )∗ . Then for any ξ ∈ D(T ) = D(xT ) we have 

   (xT )∗ η ξ = η (xT )ξ  = η x(T ξ ) = x ∗ η T ξ ,

i.e. x ∗ η ∈ D(T ∗ ) and T ∗ (x ∗ η) = (xT )∗ η. In other words η ∈ D(T ∗ x ∗ ) and (T ∗ x ∗ )η = (xT )∗ η, which means that (xT )∗ ⊂ T ∗ x ∗ . 

We also have additive analogs of Propositions 8.13 and 8.14: Proposition 8.15 (1) Let T and S be operators such that T + S is densely defined. Then T and S are densely defined and T ∗ + S ∗ ⊂ (T + S)∗ . (2) Let T be densely defined and let x ∈ B(H). Then (T + x)∗ = T ∗ + x ∗ . Proof Ad (1). Clearly if D(T + S) = D(T ) ∩ D(S) is dense in H then so are D(T ) and D(S). Let φ ∈ D(T ∗ + S ∗ ) = D(T ∗ ) ∩ D(S ∗ ). Then for any ψ ∈ D(T + S) we have     φ (T + S)ψ = φ T ψ + φ Sψ = T ∗ φ ψ + S ∗ φ ψ     = T ∗ φ + S ∗ φ ψ = (T ∗ + S ∗ )φ ψ ,  so φ ∈ D (T + S)∗ and (T + S)∗ φ = T ∗ φ + S ∗ φ.  Ad (2). We already know that (T + x)∗ ⊃ T ∗ + x ∗ . Take η ∈ D (T + x)∗ . Then the functional D(T + x) = D(T )  ψ −→ η (T + x)ψ ∈ C is continuous, and consequently so is the functional D(T )  ψ −→ η T ψ = η (T + x)ψ − η xψ ∈ C In other words η ∈ D(T ∗ ). Using this fact we can compute     η (T + x)ψ = η T ψ + η xψ = T ∗ η ψ + x ∗ η ψ     = T ∗ η + x ∗ η ψ = (T ∗ + x ∗ )η ψ , which shows that (T + x)∗ ⊂ T ∗ + x ∗ .



110

Chapter 8 • Operators and Their Graphs

8.4

Spectrum

Let T be a closed densely defined operator on H. We say that T is invertible if T is a bijection of D(T ) onto H. By the closed graph theorem the inverse bijection T −1 : H → D(T ) is then bounded. The spectrum of the operator T is defined in the same way as for bounded operators:   σ (T ) = λ ∈ C λ1 − T is not invertible . Just as in the case of bounded operators, we put ρ(T ) = C K σ (T ) and call this set the resolvent set of T . Proposition 8.16 Let T be a closed densely defined operator on H. Then σ (T ) is a closed subset of C.

8

Proof Take λ0 ∈ ρ(T ). Then for any λ ∈ C such that −1 |λ − λ0 | < (λ0 1 − T )−1 the series ∞  (λ0 − λ)n (λ0 1 − T )−n−1 n=0

converges in B(H) to a sum r. We will now check that r = (λ1 − T )−1 , i.e. (1) for ψ ∈ D(λ1 − T ) = D(T ) we have r(λ1 − T )ψ = ψ, (2) for ξ ∈ H we have rξ ∈ D(T ) and (λ1 − T )rξ = ξ . Ad (1). Let ψ ∈ D(T ). Then (λ1 − T )ψ = (λ − λ0 )ψ + (λ0 1 − T )ψ and consequently r(λ1 − T )ψ = −

∞  (λ − λ0 )n+1 (λ0 1 − T )−n−1 ψ n=0

+

∞  (λ − λ0 )n (λ0 1 − T )−n ψ = ψ. n=0

Ad (2). For any n ∈ Z+ we have (λ0 1 − T )−n−1 ξ ∈ D(T ), so putting ξN =

N  (λ0 − λ)n (λ0 1 − T )−n−1 ξ, n=0

8

111 8.4 · Spectrum

we obtain a sequence (ξN )N ∈N of elements of D(λ1 − T ) converging to rξ . Moreover  (λ1 − T )ξN = (λ − λ0 )1 + (λ0 1 − T ) ξN = (λ − λ0 )ξN +

N  (λ0 − λ)n (λ0 1 − T )−n ξ n=0

= (λ − λ0 )ξN + (λ0 − λ)

N  (λ0 − λ)n−1 (λ0 1 − T )−n ξ n=0

= (λ − λ0 )ξN  + (λ0 − λ)

1 λ0 −λ ξ

= (λ − λ0 )ξN + (λ0 − λ)



+

N −1 

(λ0 − λ) (λ0 1 − T )

k=0 1 λ0 −λ ξ

+ ξN −1

k

−k−1

 ξ



= ξ + (λ − λ0 )(ξN − ξN −1 ) −−−−→ ξ. N →∞

Thus, by closedness of (λ1 − T ), we have rξ ∈ D(λ1 − T ) and (λ1 − T )rξ = ξ . This way we showed that (λ1 − T ) is invertible, so λ ∈ ρ(T ). It follows that ρ(T ) is an open subset of C, an therefore σ (T ) = C K ρ(T ) is closed. 

It is worth mentioning that one can construct examples of closed densely defined operators T with σ (T ) = ∅, as well as operators whose spectrum is all of C (see e.g. [ReSi1 , Example 5, p. 254]). It is also not hard to see that if u is unitary, then σ (uT u∗ ) = σ (T ). Finally let us add that some authors prefer a slightly different definition of the spectrum. In this other approach the spectrum of T is considered as a subset of the Riemann sphere C and by definition contains the point ∞ whenever T is not bounded. This version of the spectrum is sometimes called the extended spectrum. One of the benefits of considering the extended spectrum is that it is always a non-empty and compact subset of C, regardless whether the considered operator is bounded or not.

Notes Fundamentals of the theory of unbounded operators on a Hilbert space presented above are developed further in monographs such as [AkGl, Kat, Mau, ReSi1 , ReSi2 ]. Moreover, many textbooks of general functional analysis have separate sections dealing with unbounded operators (e.g. [Ped, Chapter 5], [Rud2 , Chapter 13]). Our intention in this chapter was to introduce the reader to the topic reserving a more in-depth analysis of several aspects of the theory for the following chapters. Many examples and exercises can be found in textbooks and monographs mentioned above.

9

113

z-Transform © Springer Nature Switzerland AG 2018 P. Sołtan, A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-319-92061-0_9

This chapter will be devoted to developing an extremely useful tool for dealing with unbounded operators, namely the so called z-transform. It was introduced in a context much wider than the theory of operators on Hilbert spaces by S.L. Woronowicz (see [Lan, WoNa]). As we already mentioned a couple of times, the z-transform is a way to encode full information about a given closed densely defined operator T in a bounded operator zT . The procedure of passing from T to zT requires several preliminary results which will be presented in  Sect. 9.1. As an illustration of the use of the z-transform we will provide a simple proof of existence of polar decomposition of closed operators given in  Sect. 9.3.

9.1

The Operator T ∗ T

Let T be a closed operator on H. A subspace D ⊂ D(T ) is called a core for T if T is  equal to the closure of the restriction T D of T to D . From now on assume that T is closed and densely defined. Recall that 

0 1 G(T ∗ ) = G(T ) = −1 0 ⊥

It follows that for any

2ξ 3 η

0 T ∗φ −φ

1 ∗

φ ∈ D(T ) .

∈ H ⊕ H there exists ψ ∈ D(T ) and φ ∈ D(T ∗ ) such that

   ξ ψ T ∗φ = + . η Tψ −φ

(9.1)

Proposition 9.1 For any ξ ∈ H there exists a unique ψ ∈ D(T ∗ T ) such that ξ = (1 + T ∗ T )ψ. Moreover ψ ≤ ξ .

114

Chapter 9 • z-Transform

Proof Put η = 0 in (9.1). Then there exist ψ ∈ D(T ) and φ ∈ D(T ∗ ) such that    ξ ψ T ∗φ = + −φ 0 Tψ which means precisely that ξ = ψ + T ∗ φ and φ = T ψ. In other words ψ ∈ D(T ∗ T ) and ψ + T ∗T ψ = ξ . We have  

ξ 2 = ψ + T ∗ T ψ ψ + T ∗ T ψ       = ψ ψ + T ∗ T ψ ψ + ψ T ∗ T ψ + T ∗ T ψ T ∗ T ψ = ψ 2 + 2 T ψ 2 + T ∗ T ψ 2 ≥ ψ 2 .

9

The above estimate shows also that the vector ψ is uniquely determined by ξ . Indeed: if ξ = ψ + T ∗ T ψ = ψ  + T ∗ T ψ  then 0 = (ψ − ψ  ) + T ∗ T (ψ − ψ  ) and we obtain ψ − ψ  ≤ 0, so that ψ = ψ  .



It follows from Proposition 9.1 that ▬ D(T ∗ T ) = {0}, ▬ the map D(T ∗ T )  ψ −→ ψ + T ∗ T ψ ∈ H

(9.2)

is a bijection which does not decrease the norm. In particular, the map (1 + T ∗ T )−1 , i.e. the inverse of (9.2) is continuous and (1 + T ∗ T )−1 ≤ 1.

Theorem 9.2 Let T be a closed densely defined operator. Then (1) the operator T ∗ T is closed, (2) D(T ∗ T ) is a core for T .

9

115 9.2 · z-Transform of a Closed Operator

Proof   We begin with Statement (2). Suppose G T D(T ∗ T ) is not dense in G(T ). This means that 2 3   there exists a non-zero φ ∈ D(T ) such that the vector Tφφ is orthogonal to G T D(T ∗ T ) , i.e.   φ ψ ⊥ , ψ ∈ D(T ∗ T ) Tφ Tψ or, in other words, φ ψ + T φ T ψ = 0,

ψ ∈ D(T ∗ T ).

This, however, means that 

 φ ψ + T ∗ T ψ = 0,

ψ ∈ D(T ∗ T ),

so that φ ⊥ H. This contradiction shows that D(T ∗ T ) is a core for T . Statement (1) is proved as follows: the operator (1 + T ∗ T )−1 is bounded and consequently its graph is closed. Therefore the graph of 1 + T ∗ T is closed, i.e. 1 + T ∗ T is closed. Finally T ∗ T = (1 + T ∗ T ) + (−1) is closed by Proposition 8.10. 

Corollary 9.3 Let T be a closed densely defined operator on H. Then D(T ∗ T ) is dense in H. Proof A core of an operator is dense in its domain and T is assumed to be densely defined.

9.2



z-Transform of a Closed Operator

Throughout this section T will be a closed densely defined operator on H. Lemma 9.4 The operator (1 + T ∗ T )−1 is positive. Proof Take ξ ∈ H and let ψ ∈ D(T ∗ T ) be such that ξ = ψ + T ∗ T ψ. Then 

   ξ (1 + T ∗ T )−1 ξ = ψ + T ∗ T ψ (1 + T ∗ T )−1 (1 + T ∗ T )ψ     = ψ + T ∗ T ψ ψ = ψ ψ + T ∗ T ψ ψ = ψ 2 + T ψ 2 ≥ 0. 

116

Chapter 9 • z-Transform

Lemma 9.4 allows us to consider the operator (1 + T ∗ T )− 2 , i.e. the positive square root of the positive operator (1 + T ∗ T )−1 . Note that its range is dense in H, as 1

(1 + T ∗ T )− 2 H ⊃ (1 + T ∗ T )− 2 (1 + T ∗ T )− 2 H 1

1

1

= (1 + T ∗ T )−1 H = D(T ∗ T ) and the operator T ∗ T is densely defined.

Theorem 9.5 Let T be a closed densely defined operator on H. Then 1 (1) (1 + T ∗ T )− 2 H = D(T ), 1 1 (2) T (1 + T ∗ T )− 2 ∈ B(H) and T (1 + T ∗ T )− 2 ≤ 1.

9

Proof Take η ∈ H. Then (1 + T ∗ T )−1 η ∈ D(T ∗ T ) ⊂ D(T ) and   T (1 + T ∗ T )−1 η 2 = T (1 + T ∗ T )−1 η T (1 + T ∗ T )−1 η   = (1 + T ∗ T )−1 η T ∗ T (1 + T ∗ T )−1 η   ≤ (1 + T ∗ T )−1 η (1 + T ∗ T )(1 + T ∗ T )−1 η   = (1 + T ∗ T )−1 η η   1 1 1 2 = (1 + T ∗ T )− 2 η (1 + T ∗ T )− 2 η = (1 + T ∗ T )− 2 η . 1

Therefore, for vectors ξ of the form (1 + T ∗ T )− 2 η we have T (1 + T ∗ T )− 12 ξ ≤ ξ .

(9.3)

Now let ζ ∈ H. Then there exists a sequence (ξn )n∈N of elements of the subspace (1 + 1 T ∗ T )− 2 H converging to ζ . It follows that 1

1

(1 + T ∗ T )− 2 ξn −−−→ (1 + T ∗ T )− 2 ζ. n→∞

Moreover, thanks to the estimate (9.3), we also have T (1 + T ∗ T )− 12 ξn − T (1 + T ∗ T )− 12 ξm 1 = T (1 + T ∗ T )− 2 (ξn − ξm ) ≤ ξn − ξm ,

9

117 9.2 · z-Transform of a Closed Operator

 1 T (1 + T ∗ T )− 2 ξn n∈N converges. This way, form

which implies that the sequence

1

closedness of T we infer that (1 + T ∗ T )− 2 ζ ∈ D(T ) and consequently 1

(1 + T ∗ T )− 2 H ⊂ D(T ). 1 By Corollary 8.12, this shows that T (1 + T ∗ T )− 2 ∈ B(H) and (9.3) gives T (1 + 1 T ∗ T )− 2 ≤ 1. Statement (2) is proved. 1

To finish the proof of (1) we have to show that D(T ) ⊂ (1 + T ∗ T )− 2 H. Let us take any ξ ∈ D(T ). Since D(T ∗ T ) is a core for T , there exists a sequence (ψn )n∈N of elements of D(T ∗ T ) such that ψn −−−→ ξ n→∞

and

T ψn −−−→ T ξ. n→∞

 1 Put φn = (1 + T ∗ T )ψn . Then the sequence (1 + T ∗ T )− 2 φn n∈N converges, as (1 + T ∗ T )− 12 (φn − φm ) 2   1 1 = (1 + T ∗ T )− 2 (φn − φm ) (1 + T ∗ T )− 2 (φn − φm )   = φn − φm (1 + T ∗ T )−1 (φn − φm )   = (1 + T ∗ T )(ψn − ψm ) ψn − ψm = ψn − ψm 2 + T ψn − T ψm 2 −−−−−→ 0. n,m→∞

It follows that 1

1

ξ = lim ψn = lim (1 + T ∗ T )− 2 (1 + T ∗ T )− 2 φn n→∞

n→∞

1

1

1

= (1 + T ∗ T )− 2 lim (1 + T ∗ T )− 2 φn ∈ (1 + T ∗ T )− 2 H. n→∞



The bounded operator T (1 + T ∗ T )− 2 is called the z-transform of T and is denoted by the symbol zT . Let us note once more that for any closed densely defined T we have

zT ≤ 1. In particular zT ∗ zT ≤ 1, so that the operator 1 − zT ∗ zT is positive. The next theorem makes precise the claim that all information about T is contained in zT . 1

Chapter 9 • z-Transform

118

Theorem 9.6 Let T be a closed densely defined operator on H. Then 0 1 (1 − zT ∗ zT ) 2 ξ G(T ) = zT ξ

1 ξ ∈H .

Proof We have 0 G(T ) = 0

9

=

ψ ∈ D(T ) 1

1

(1 − T ∗ T )− 2 ξ 1

0 =

1 ψ Tψ

T (1 − T ∗ T )− 2 ξ

ξ ∈H 1

1

(1 − T ∗ T )− 2 ξ zT ξ

(9.4)

ξ ∈H .

Now, remembering that for any ξ ∈ H the vector (1 + T ∗ T )−1 ξ belongs to D(T ∗ T ), we compute   (1 + T ∗ T )− 12 ξ 2 = (1 + T ∗ T )− 12 ξ (1 + T ∗ T )− 12 ξ     = ξ (1 + T ∗ T )−1 ξ = (1 + T ∗ T )−1 ξ ξ   = (1 + T ∗ T )−1 ξ (1 + T ∗ T )(1 + T ∗ T )−1 ξ  2  = (1 + T ∗ T )−1 ξ + (1 + T ∗ T )−1 ξ T ∗ T (1 + T ∗ T )−1 ξ 2 2 = (1 + T ∗ T )−1 ξ + T (1 + T ∗ T )−1 ξ 1 1 2 = (1 + T ∗ T )− 2 (1 + T ∗ T )− 2 ξ 1 1 2 + T (1 + T ∗ T )− 2 (1 + T ∗ T )− 2 ξ . 1

Therefore, setting ψ = (1 + T ∗ T )− 2 ξ we get 2 1

ψ 2 = (1 + T ∗ T )− 2 ψ + zT ψ 2 .

(9.5)

Such vectors ψ form a dense subset D(T ) of H, an consequently (9.5) holds for all ψ ∈ H.

9

119 9.2 · z-Transform of a Closed Operator

It follows from (9.5) and the polarization formula that     φ ψ = φ (1 + T ∗ T )−1 ψ + φ zT ∗ zT ψ ,

ψ, φ ∈ H.

This, in turn, means that ψ = (1 + T ∗ T )−1 ψ + zT ∗ zT ψ,

ψ ∈ H,

so that (1 + T ∗ T )−1 = 1 − zT ∗ zT . 1

(9.6) 1

Therefore (1 + T ∗ T )− 2 = (1 − zT ∗ zT ) 2 which, together with (9.4), yields 0 1 (1 − zT ∗ zT ) 2 ξ G(T ) = zT ξ

1 ξ ∈H . 

Corollary 9.7 Let S and T be closed densely defined operators on H. If zS = zT then S = T.

Let us note here that if T is a closed densely defined operator then zT satisfies ker(1 − zT ∗ zT ) = {0}.  ⊥ Indeed: ker(1 − zT ∗ zT ) = (1 − zT ∗ zT )H and by (9.6) we have (1 − zT ∗ zT )H = (1 + T ∗ T )−1 H = D(T ∗ T ), ⊥  so that (1 − zT ∗ zT )H = {0}. Lemma 9.8 Let z ∈ B(H) be such that z ≤ 1. Then for any f ∈ C([0, 1]) we have f (z∗ z)z∗ = z∗ f (zz∗ )

and

zf (z∗ z) = f (zz∗ )z.

Proof We have σ (z∗ z), σ (zz∗ ) ⊂ [0, 1]. Let (fn )n∈N be a sequence of polynomials converging uniformly to f on [0, 1]. Then fn (z∗ z)z∗ = z∗ fn (zz∗ ),

n∈N

Chapter 9 • z-Transform

120

and fn (z∗ z)z∗ −−−→ f (z∗ z)z∗ , n→∞

z∗ fn (zz∗ ) −−−→ z∗ f (zz∗ ). n→∞

It follows that f (z∗ z)z∗ = z∗ f (zz∗ ). The second formula is obtained from the first one for f by applying the involution. 

Theorem 9.9 An operator z ∈ B(H) is a z-transform of a closed densely defined operator T if and only if z satisfies ker(1 − z∗ z) = {0}.

z ≤ 1 and

9

Proof We checked above that the conditions of the theorem are necessary for z to be a z-transform of a closed densely defined operator. Assume now that z ∈ B(H) satisfies z ≤ 1 and ker(1 − z∗ z) = {0}. Put 0 G=

1

(1 − z∗ z) 2 ξ zξ

1 ξ ∈H .

We will check that G is a graph of a closed densely defined operator (see Corollary 8.4). To that end let us define  1 (1 − z∗ z) 2 −z∗ Uz = ∈ B(H ⊕ H). 1 z (1 − zz∗ ) 2 Then Uz is unitary: Uz∗ Uz

  1 1 (1 − z∗ z) 2 (1 − z∗ z) 2 z∗ −z∗ = 1 1 ∗ −z (1 − zz ) 2 z (1 − zz∗ ) 2  1 1 (1 − z∗ z) + z∗ z z∗ (1 − zz∗ ) 2 − (1 − z∗ z) 2 z∗ = 1 1 (1 − zz∗ ) 2 z − z(1 − z∗ z) 2 zz∗ + (1 − zz∗ )  1 0 = 0 1

121 9.2 · z-Transform of a Closed Operator



by Lemma 9.8 applied to the function f (t) = Uz Uz∗

1 − t. Similarly

  1 1 (1 − z∗ z) 2 (1 − z∗ z) 2 −z∗ z∗ = 1 1 ∗ 2 z (1 − zz ) −z (1 − zz∗ ) 2  1 1 (1 − z∗ z) + z∗ z (1 − z∗ z) 2 z∗ − z∗ (1 − zz∗ ) 2 = 1 1 z(1 − z∗ z) 2 − (1 − zz∗ ) 2 z zz∗ + (1 − zz∗ )  1 0 = . 0 1

Note further that 0 1 ξ G = Uz ξ ∈H , 0 so that G is a closed subspace of H ⊕ H (as the image of one under a unitary map). We check 2 3 now that G does not contain non-zero vertical vectors: suppose φ0 ∈ G. Then there exists ξ ∈ H such that 

 1 0 (1 − z∗ z) 2 ξ = . φ zξ 1

This means that (1−z∗ z) 2 ξ = 0, but this implies (1−z∗ z)ξ = 0, so ξ ∈ ker(1−z∗ z) = {0}. To see that G⊥ does not contain non-zero horizontal vectors we note that 0





G = Uz  0 = Uz

1

ξ 0

ξ ∈H

1⊥ 

ξ 0

ξ ∈H

0 =

−z∗ η 1 (1 − zz∗ ) 2 η

Suppose that 



2φ3 0

0 = Uz

0 η

1 η∈H

1 η∈H .

∈ G⊥ . Then there exists η ∈ H such that

 −z∗ η φ . = 1 0 (1 − zz∗ ) 2 η 1

Therefore (1 − zz∗ ) 2 η = 0, so also (1 − zz∗ )η = 0. In other words zz∗ η = η.

9

Chapter 9 • z-Transform

122

Multiplying this equality from the left by z∗ gives z∗ η ∈ ker(1 − z∗ z) = {0}, so φ = − z∗ η = 0.



Repeating the reasoning of the proof of Theorem 9.9 we obtain the following fact: for a closed densely defined operator T on H we have 0

1

−zT ∗ ξ 1 (1 − zT zT ∗ ) 2 ξ



G(T ) =

ξ ∈H .

(9.7)

Indeed: 0 ψ Tψ

G(T ) =

1

 ψ ∈ D(T ) = UzT H ⊕ {0} ,

where

9

UzT

 1 −zT ∗ (1 − zT ∗ zT ) 2 , = 1 zT (1 − zT zT ∗ ) 2

 2 ξ 3 and H ⊕ {0} = ξ ∈ H . Since UzT is unitary (the argument for that is the 0 calculation from the proof of Theorem 9.9), we obtain



⊥ ⊥   = UzT {0} ⊕ H , = UzT H ⊕ {0} G(T )⊥ = UzT H ⊕ {0} where {0} ⊕ H denotes

2 0 3 η

 ξ ∈ H . This immediately gives (9.7).

Proposition 9.10 Let T be closed densely defined operator on H. Then zT ∗ = zT ∗ . Proof Using formula (9.7) we get 



0 1 G(T )⊥ −1 0  0 1 −zT ∗ ξ 0 1 = ξ ∈ H 1 −1 0 (1 − zT zT ∗ ) 2 ξ 0 1 1 (1 − zT zT ∗ ) 2 ξ = ξ ∈H . zT ∗ ξ

G(T ) =

It follows that zT ∗ is the z-transform of T ∗ .



We end this section with a remark on how the z-transform behaves under unitary equivalence:

123 9.3 · Polar Decomposition

ⓘ Remark 9.11 Let H and K be Hilbert spaces and let T be a closed densely defined operator on H. Let u ∈ B(H, K) be a unitary operator and let S = u∗ T u, i.e.     D(S) = ψ ∈ K uψ ∈ D(T ) = u∗ η η ∈ D(T ) and S(u∗ η) = u∗ T η for all η ∈ D(T ). Then S is a closed and densely defined operator on K and zS = u∗ zT u. Indeed: 1 ψ ψ ∈ D(S) G(S) = Sψ 1 0 u∗ η η ∈ D(T ) = u∗ T η 0 1 1 u∗ (1 − zT ∗ zT ) 2 ξ = ξ ∈H u∗ zT ξ 0 1 1 u∗ (1 − zT ∗ zT ) 2 uφ = φ ∈ K u∗ zT uφ 0 1 1 (1 − z∗ z) 2 φ = φ∈K zφ 0

with z = u∗ zT u. In view of Corollary 9.7, this shows that zS = u∗ zT u.

9.3

Polar Decomposition

An operator T on H is called positive if ξ T ξ  ≥ 0,

ξ ∈ D(T ).

Unlike in the case of bounded operators, positivity of an unbounded operator does not imply its self-adjointness. However, for a closed densely defined operator T the operator T ∗ T is always positive and self-adjoint.1

is easy to see that T ∗ T is positive. To see that T ∗ T is self-adjoint let a = (1 + T ∗ T )−1 . Then a is self2 0 13 2 0 13 ⊥ adjoint, so −1 0 G(a) = G(a). Applying the unitary operator 1 0 to both sides of this equality we obtain 1 It



0 −1 G(1 + T ∗ T )⊥ = G(1 + T ∗ T ), 1 0

which shows that S = 1 + T ∗ T is self-adjoint. Thus, by Proposition 8.15(2) we get (T ∗ T )∗ = (S − 1)∗ = S − 1 = T ∗T .

9

124

Chapter 9 • z-Transform

Lemma 9.12 An operator T is positive if and only if 1

(1 − zT ∗ zT ) 2 zT ≥ 0. In particular T is positive and self-adjoint if and only if zT ≥ 0. Proof 1 A vector ψ belongs to D(T ) if and only if ψ is of the form (1 − zT ∗ zT ) 2 ξ for some ξ ∈ H. Therefore   1 1 ψ T ψ = (1 − zT ∗ zT ) 2 ξ T (1 − zT ∗ zT ) 2 ξ     1 1 = (1 − zT ∗ zT ) 2 ξ zT ξ = ξ (1 − zT ∗ zT ) 2 zT ξ which proves the first part of the lemma. Furthermore, T = T ∗ if and only if zT = zT ∗ , so T is positive and self-adjoint if and only if zT = zT ∗ and

9



 1 1 (1 − zT ∗ zT ) 4 η zT (1 − zT ∗ zT ) 4 η   1 = η (1 − zT ∗ zT ) 2 zT η ≥ 0,

η ∈ H.

(9.8)

1

It is easy to see that the range of (1 − zT ∗ zT ) 4 is dense in H, and hence the condition (9.8) is equivalent to ϕ zT ϕ ≥ 0,

ϕ ∈ H,

i.e. zT ≥ 0.



Theorem 9.13 Let T be a closed densely defined operator on H. Then there exists a unique pair of operators (u, K) such that (1) T = uK, (2) K is positive and self-adjoint, (3) u∗ u is the projection onto the closure of the range of K.

Proof Let zT = u|zT | be the polar decomposition of the bounded operator zT . Since |zT | =

zT ≤ 1 and  ker 1 − |zT |∗ |zT | = ker(1 − zT ∗ zT ),

9

125 9.3 · Polar Decomposition

the operator |zT | is the z-transform of some closed densely defined operator K: |zT | = zK . As |zT | is positive, we have K = K ∗ and K is positive. Also D(K) = D(T ), because 1

1

1

D(K) = (1 − zK ∗ zK ) 2 H = (1 − |zT |∗ |zT |) 2 H = (1 − zT ∗ zT ) 2 H = D(T ). Moreover 0 G(K) = 0 =

1

1

(1 − zK ∗ zK ) 2 ξ zK ξ ∗z

1 2

(1 − zT T ) ξ |zT |ξ

ξ ∈H (9.9)

1 ξ ∈H

and obviously 0 1 (1 − zT ∗ zT ) 2 ξ G(T ) = zT ξ It follows that  G(T ) =

1 ξ ∈H .

10 G(K), 0 u

i.e. T = uK. Finally u∗ u is the projection onto the closure of the range of |zT | and formula (9.9) shows that the latter coincides with the range of K. Let us now address the uniqueness of the pair (u, K). Let (v, D) be another pair such that ▬ T = vD, ▬ D is positive and self-adjoint, ▬ v ∗ v is the projection onto the closure of the range of D. Then, on one hand, using Proposition 8.14 we obtain T ∗ T = (uK)∗ (uK) = Ku∗ uK = K 2 , and on the other hand T ∗ T = (vD)∗ (vD) = Dv ∗ vD = D 2 . It follows that 1

1

1

zT = T (1 + T ∗ T )− 2 = T (1 + K 2 )− 2 = T (1 + D 2 )− 2

Chapter 9 • z-Transform

126

and consequently 1

1

uK(1 + K 2 )− 2 = vD(1 + D 2 )− 2 or, in other words, uzK = vzD . Applying the operation x → x ∗ x to both sides of this equality yields zK u∗ uzK = zD v ∗ vzD . Since the range of the z-transform coincides with the range of the operator, we have u∗ uzK = zK and v ∗ vzD = zD and therefore zK 2 = zD 2 .

9

The operators zK and zD are positive, so by uniqueness of positive square roots we get zK = zD and thus K = D. The equality u = v follows now from uK = vK and the fact that u∗ u = v ∗ v is the projection onto the closure of the range of K in the same way as for bounded operators. 

The decomposition T = uK of a closed densely defined operator T obtained in Theorem 9.13 is called the polar decomposition of T . The partial isometry u is sometimes referred to as the phase of T , while the positive self-adjoint operator K is called the modulus of absolute value of T and is denoted by the symbol |T |.

Notes The concept of the z-transform was introduced for the first time in a framework far more general than the theory of operators on Hilbert spaces. Given any C∗ -algebra A consider a vector space E endowed with a right action of A and a “scalar product” with values in A. The latter is a sesquilinear map · · : E × E → A possessing several properties which guarantee that · · defines a norm on E in a way analogous to how it is defined on spaces with a genuine scalar product:

ξ E =

ξ ξ  ,

ξ ∈E

(for details see [Lan, Chapter 1]). If E is complete in this norm then it is called a Hilbert C∗ -module over A. In particular, Hilbert C∗ -modules over A = C are nothing else than Hilbert spaces. It turns out that many problems of pure mathematics require the additional generality of Hilbert C∗ -modules over non-trivial C∗ -algebras instead of Hilbert spaces. A surprising feature of Hilbert C∗ -modules is that not all linear operators on such a module necessarily have an adjoint, even bounded ones. The situation becomes even

127 9.3 · Polar Decomposition

more complicated when we analyze unbounded operators. The notion of z-transform turned out to be crucial for the development of the theory of operators on Hilbert C∗ modules. This theory is described very thoroughly in the book [Lan]. Many interesting analogies and differences between studying operators on Hilbert spaces and on Hilbert C∗ -modules are highlighted in the paper [WoNa].

9

129

Spectral Theorems © Springer Nature Switzerland AG 2018 P. Sołtan, A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-319-92061-0_10

Just as in the case of bounded operators, spectral theorem for unbounded operators assumes various forms. We will begin with continuous functional calculus which can be defined exclusively using the z-transform. Next we will move on to Borel functional calculus and finally assign to each self-adjoint operator its spectral measure and discuss functional calculus for unbounded functions.

10.1 Continuous Functional Calculus Let T be a self-adjoint operator on H. We will begin by defining functional calculus for bounded continuous functions. The space Cb (R) of bounded continuous functions on R is a C∗ -algebra with natural operations of addition and multiplications, the uniform norm · ∞ and complex conjugation of functions as involution. Functional calculus should be a unital ∗-homomorphism Cb (R) → B(H). Moreover, putting ζ (t) =

√ t , 1+t 2

t ∈R

we define a continuous and bounded function on R (in fact ζ is a homeomorphism R → ]−1, 1[) and we would like to have ζ (T ) = zT . Since zT is a self-adjoint (bounded) operator, we can consider the map Cb (R)  f −→ (f ◦ ζ −1 )(zT ) which formally would satisfy our requirements. However, the function f ◦ ζ −1 belongs to Cb (]−1, 1[) while σ (zT ) ⊂ [−1, 1]1 and hence the construction of continuous functional calculus for T will have to be more involved.

1 Moreover,

one can show that if σ (zT ) ⊂ ]−1, 1[ then T is bounded.

10

130

Chapter 10 • Spectral Theorems

Theorem 10.1 Let T be a self-adjoint operator. Then there exists a unique unital ∗-homomorphism Cb (R) → B(H) denoted by Cb (R)  f −→ f (T ) ∈ B(H) such that ζ (T ) = zT .

Proof For any f ∈ Cb (R) the function  1 ]−1, 1[  t −→ f ζ −1 (t) (1 − t 2 ) 2

10

extends to a continuous function on [−1, 1]. Let us denote this extension by f. Now take 1 1 ξ ∈ D(T ). Then there exists a unique η ∈ H such that ξ = (1+T ∗ T )− 2 η = (1−zT ∗ zT ) 2 η (see Eq. (9.6)). Consider the map D(T )  ξ −→ f(zT )η ∈ H.

(10.1)

We will show that it is bounded. Indeed: the function f satisfies 

 2 ff (t) = f(t) ≤ f 2∞ (1 − t 2 ),

t ∈ [−1, 1].

Thus, using properties of continuous functional calculus for bounded self-adjoint operators, we obtain f(zT )∗ f(zT ) ≤ f 2∞ (1 − zT ∗ zT ). Therefore     f(zT )η 2 = f(zT )η f(zT )η = η f(zT )∗ f(zT )η   ≤ f 2∞ η (1 − zT ∗ zT )η   1 1 = f 2∞ (1 − zT ∗ zT ) 2 η (1 − zT ∗ zT ) 2 η = f 2∞ ξ 2 . It follows that the map (10.1) extends to a bounded operator H → H with norm not larger than f ∞ . Let us denote it by the symbol f (T ). Clearly the map f → f (T ) is linear and if f is constant and equal to 1, then f (T ) = 1. We will now show that f (T )g(T ) = (fg)(T ). To that end note that for any h ∈ Cb (R) we have (by definition) h(T )(1 − zT ∗ zT ) 2 φ =  h(zT )φ, 1

φ ∈ H,

(10.2)

131 10.1 · Continuous Functional Calculus

so that for any ξ ∈ D(T ∗ T ) = (1 − zT ∗ zT )H f (T )g(T )ξ = f (T )g(zT )(1 − zT ∗ zT )φ 1

1

= f (T )g(T )(1 − zT ∗ zT ) 2 (1 − zT ∗ zT ) 2 φ 1

= f (T ) g (zT )(1 − zT ∗ zT ) 2 φ = f (T )(1 − zT ∗ zT ) 2  g (zT )φ = f(zT ) g (zT )φ, 1

where φ is the vector satisfying ξ = (1 − zT ∗ zT )φ. Furthermore, for any t ∈ ]−1, 1[  1  1 f(t) g(t) = f ζ −1 (t) (1 − t 2 ) 2 g ζ −1 (t) (1 − t 2 ) 2  1 1 1 = (fg) ζ −1 (t) (1 − t 2 ) 2 (1 − t 2 ) 2 = (f4 g)(t)(1 − t 2 ) 2 , and so f (T )g(T )ξ = (f g )(zT )(1 − zT ∗ zT ) 2 φ 1

1

1

= (fg)(T )(1 − zT ∗ zT ) 2 (1 − zT ∗ zT ) 2 φ = (fg)(T )(1 − zT ∗ zT )φ = (fg)(T )ξ. This proves that f (T )g(T ) = (fg)(T ), because D(T ∗ T ) is dense in H. Finally the equality f (T ) = f (T )∗ follows directly from the definition of the operator f (T ). We have shown that f → f (T ) is a unital ∗-homomorphism from Cb (R) to B(H). Of 1 1 course, for f = ζ we have f(t) = t (1 − t 2 ) 2 , so that for ξ = (1 − zT ∗ zT ) 2 η ∈ D(T ) we get f (T )ξ = f(zT )η = zT (1 − zT ∗ zT ) 2 φ = zT ξ 1

which means that f (T ) = zT . Let us now address the uniqueness of a homomorphism satisfying the condition of the theorem. Let  : Cb (R) → B(H) be a unital ∗-homomorphism such that (ζ ) = zT . Then the values of  are uniquely determined on polynomials in ζ . Moreover, we know from Proposition 4.9 ( Sect. 4.2) that any unital ∗-homomorphism from a Banach ∗-algebra to B(H) is continuous, so that the values of  are also uniquely determined on all functions of the form ϕ ◦ ζ with ϕ ∈ C([−1, 1]). In other words, the condition that (ζ ) = zT uniquely determines values of  on all functions from Cb (R) possessing limits at ±∞. We will see that this already determines the homomorphism  uniquely on all of Cb (R). For f ∈ Cb (R) define ⎧ ⎪ ⎪ ⎨f (−n), fn (t) = f (t), ⎪ ⎪ ⎩ f (n),

t < −n, −n ≤ t ≤ n, t > n,

t ∈ R.

10

132

Chapter 10 • Spectral Theorems

We will prove that (fn )ξ −−−→ (f )ξ,

ξ ∈ D(T )

n→∞

(10.3)

which guarantees that the operator (f ) is uniquely determined on a dense subset of H by values of  on functions which have limits at ±∞. This, in turn, gives us uniqueness of continuous functional calculus21  Let g(t) = 1 − ζ (t)2 2 . Since values of  on polynomials in zT are uniquely determined, we obtain (g 2 ) = 1 − zT ∗ zT and thus, by uniqueness of positive square roots, we get 1

(g) = (1 − zT ∗ zT ) 2 . 1

Now for ξ ∈ D(T ) there exists a unique η ∈ H such that ξ = (1 − zT ∗ zT ) 2 η, i.e.

10

ξ = (g)η and it follows that (fn )ξ = (fn )(g)η = (fn g)η. Finally we have fn g −−−→ fg uniformly on R, because n→∞

  (fn g)(t) − (fg)(t) ≤ 2 f ∞ sup g(t) −−−→ 0, |t|>n

n→∞

and consequently (fn g) −−−→ (fg) in B(H). Therefore n→∞

(fn )ξ = (fn g)η −−−→ (fg)η = (f )ξ n→∞



and (10.3) is proved.

2 Clearly,

since (fn ) < C for some constant C, we also have

(fn )ξ −−−→ (f )ξ, n→∞

ξ ∈ H.

Indeed: fix ξ ∈ H and for ε > 0 choose ξ  ∈ D(T ) such that ξ − ξ  <

ε 3C .

Then

(fn )ξ − (f )ξ ≤ (fn )ξ − (fn )ξ  + (fn )ξ  − (f )ξ  + (f )ξ  − (f )ξ 2ε ≤ C ξ − ξ  + (fn )ξ  − (f )ξ  + C ξ − ξ  < 3M + (fn )ξ  − (f )ξ  .   Therefore, for n large enough, so that (fn )ξ − (f )ξ < (f )ξ < ε.

ε 3C ,

the above estimate gives (fn )ξ −

10

133 10.1 · Continuous Functional Calculus

Lemma 10.2 Let f ∈ Cb (R) be real-valued and such that f (t) = 0 for all t ∈ R. Then ker f (T ) = {0}. Proof Denote a = f (T ),

b = f(zT ),

1

c = (1 − zT ∗ zT ) 2 .

Let us first see that ker b = {0}. For this we can assume that zT is an operator of multiplication by a function on a Hilbert space L2 (, μ), i.e. zT = Mg . Furthermore, since ker(1 − zT 2 ) =   ker(1 − zT ∗ zT ) = {0}, we have μ ω g(ω) = ±1 = 0 (Proposition 4.4). The operator b = f(zT ) is then the operator of multiplication by f◦ g. As f(t) = 0 for t ∈ ]−1, 1[, we get   μ ω (f◦ g)(ω) = 0 = 0 and it follows that ker b = {0} (again by Proposition 4.4). By definition of functional calculus for T we have acη = bη,

η ∈ H,

i.e. ac = b. The operators a, b and c are self-adjoint, so ca = (ac)∗ = b∗ = b = ac. Thus b = ac = ca and consequently ker a ⊂ ker b = {0}.



Lemma 10.2 makes it simple to extend functional calculus of bounded continuous functions to all continuous functions on R. For simplicity we will restrict attention to real-valued functions. This will also make it easier to make a connection with Borel functional calculus which will be presented in the next section.

Theorem 10.3 Let T be a self-adjoint operator. Then for any real-valued f ∈ C(R) there exists a unique closed densely defined operator f (T ) such that zf (T ) = (ζ ◦ f )(T ). Moreover f (T ) is self-adjoint.

Proof Let z = (ζ ◦ f )(T ). We have z ≤ ζ ∞ = 1 and ker(1 − z∗ z) = {0} by Lemma 10.2, as 1 − z∗ z = g(T ),

134

Chapter 10 • Spectral Theorems

 2 where g(t) = 1 − ζ f (t) > 0 for all t ∈ R. It follows that z is a z-transform of a unique closed densely defined operator which we call f (T ). This operator is self-adjoint, because z = z∗ . 

10.2 Borel Functional Calculus Let T be a self-adjoint operator on H. In view of Theorem 9.10, the z-transform zT of T is a self-adjoint operator with spectrum contained in [−1, 1]. In particular zT is unitarily equivalent to a multiplication operator (Theorem 4.7). This allows us to conclude the following:

10

Theorem 10.4 Let T be a self-adjoint operator on H. Then there exist a semifinite measure space (, μ), a measurable real-valued function f on  and a unitary operator u ∈  B L2 (, μ), H such that T = uMf u∗ , i.e.   (1) D(T ) = uψ ψ ∈ L2 (, μ), f ψ ∈ L2 (, μ) , (2) for uψ ∈ D(T ) we have T uψ = uf ψ. Moreover σ (T ) = Vess (f ).

Proof Since zT is a self-adjoint operator, by Theorem 4.7 there exits a semifinite measure space (, μ), a measurable real-valued function F on  and a unitary operator u ∈  B L2 (, μ), H such that zT = uMF u∗ . Consider the operator S = u∗ T u. Then by Remark 9.11 we have zS = u∗ zT u = MF . It follows that   1 D(S) = (1 − zS ∗ zS ) 2 φ φ ∈ L2 (, μ)   1 = (1 − MF 2 ) 2 φ φ ∈ L2 (, μ)   = M√ 2 φ φ ∈ L2 (, μ) . 1−F

√ Note that since ker(1 − zS ∗ zS ) = {0}, by Proposition 4.4 we have 1 − F 2 = 0 almost everywhere on . It follows that the function f = √ F 2 is finite almost everywhere. 1−F √ Moreover, for ψ = 1 − F 2 φ ∈ D(S) we have Sψ = F φ = f ψ, so S ⊂ Mf . Both S and Mf are self-adjoint, so by Remark 8.9 they are equal. It follows that T = uMf u∗ . Let us analyze the spectrum of T . We know that σ (T ) = σ (u∗ T u) = σ (Mf ), so the theorem will be proved when we show that σ (Mf ) = Vess (f ). Let λ ∈ C K Vess (f ). Then

10

135 10.2 · Borel Functional Calculus

there exits r > 0 such that   ω |λ − f (ω)| < r is of measure zero. Therefore the function g =

1 λ−f

belongs to L∞ (, μ) and satisfies

g ∞ ≤ In particular the operator Mg is bounded. Furthermore ▬ for ϕ ∈ D(Mf ) we have (λ1 − Mf )ϕ = (λ − f )ϕ and this immediately shows that Mg (λ1 − Mf )ϕ = ϕ, ▬ for ψ ∈ L2 (, μ) we have Mg ψ ∈ D(Mf ), because 1 r.

 f   f −λ+λ   1  =  = λ  |fg| =  λ−f λ−f λ−f − 1 ≤ |λ||g| + 1,  and so |fgψ| ≤ |λ||g| + 1 |ψ|, i.e. fgψ ∈ L2 (, μ). It follows that Mg = (λ1 − Mf )−1 . Conversely, if λ ∈ Vess (f ) then for any n ∈ N the set   ω |λ − f (ω)| ≤ n1 is of strictly positive measure and consequently it contains a subset n of strictly positive 1 and finite measure. Putting ψn = √μ( χ we obtain a sequence of norm 1 vectors such ) n n

that ψn ∈ D(Mf ) for all n, as |f | ≤ |λ| +  

|f ψn |2 dμ =

1 μ(n )



1 n

on n , and thus

 2 |f |2 dμ ≤ |λ| + n1 < +∞.

n

It is easy to see that (λ1−Mf )ψn 2 ≤ n1 and hence there does not exist a bounded operator a such that a(λ1 − Mf ) = 1, i.e. λ ∈ σ (Mf ). 

As Theorem 10.4 says, any self-adjoint operator T is unitarily equivalent to an operator of multiplication by a real-valued measurable function. In particular σ (T ) is a non-empty subset of R. As in the case of bounded operators, Theorem 10.4 allows us to extend functional calculus for a self-adjoint operator from bounded continuous functions to bounded Borel functions on R. Theorem 10.5 Let T be a self-adjoint operator on H and let B (R) denote the set of bounded Borel functions on R. Then there exists a unique unital ∗-homomorphism B (R) → B(H) denoted by B (R)  g −→ g(T ) ∈ B(H)

(Continued )

136

Chapter 10 • Spectral Theorems

Theorem 10.5 (continued) such that ▬ ζ (T ) = zT , ▬ if (gn )n∈N is a uniformly bounded sequence of Borel functions converging pointwise to g then gn (T ) −−−→ g(T ) in strong topology. n→∞

Proof Just as in the proof of the analogous statement for bounded operators, we use the fact that T is unitarily equivalent to an operator of multiplication by a real-valued measurable function on a Hilbert space L2 (, μ) for some semifinite measure space (, μ). In fact one can assume that  is a locally compact topological space, μ is a Borel measure and f is Borel. Then for any bounded Borel function g on R the function g ◦ f is Borel and bounded on . For such g we define g(T ) as uMg◦f u∗ . It follows immediately from this definition that g → g(T ) is a contractive unital ∗homomorphism and that

10

  g(T ) = Mg◦f = sup |λ| λ ∈ Vess (g ◦ f ) ≤ g ∞ . Furthermore, for g = ζ we have g(T ) = u∗ M √ f

u, i.e. g(T ) = zT (cf. proof of

1+f 2

Theorem 10.4 and Remark 9.11). Finally, if (gn )n∈N is a bounded sequence of Borel functions converging pointwise to g and ξ ∈ H then gn (x)ξ − g(x)ξ 2 = uM(g −g)◦f u∗ ξ 2 n 2 = M(gn −g)◦f u∗ ξ   2  2  = gn f (ω) − g f (ω)  (u∗ ξ )(ω) dμ(ω) −−−→ 0. 

n→∞

by the dominated convergence theorem. The proof of uniqueness of the homomorphism  B σ (x)  f −→ f (x) ∈ B(H) follows the lines of the proof of the corresponding statement for bounded operators (Theorem 4.10). We know that a unital ∗-homomorphism Cb (R) → B(H) mapping ζ to zT is unique, so any unital ∗-homomorphism  : B (R) → B(H) satisfying (ζ ) = zT must coincide with the continuous functional calculus for T on Cb (R). Using this, just as in the proof of Theorem 4.10, we show that the family   L =  ∈ M (χ ) = χ (x)

137 10.3 · Spectral Measure

(1) contains all open sets (in particular  ∈ L), (2) is closed with respect to taking complements, (3) is closed with respect to countable unions of pairwise disjoint elements, i.e. L is a λ-system (see Appendix A.2). Since the family of all open subsets of R is a πsystem, by Dynkin’s theorem on π- and λ-systems (Theorem A.2), the σ -algebra of all Borel subsets of R is contained in L. Now any bounded Borel function is a pointwise limit of a bounded sequence of simple Borel functions, so the homomorphisms  and f → f (x) must  agree on all of B R . 

10.3 Spectral Measure Let T be a self-adjoint operator on H. Our aim in this section is to express T as an integral of the function λ → λ with respect to certain spectral measure on R. So far we have discussed such integrals only for bounded functions ( Sect. 4.3) and now we will deal with arbitrary Borel functions R → C. We will freely use concepts and notation introduced in  Sect. 4.3. Let E be a spectral measure on a measurable space (, M). Lemma 10.6 For any ϕ, ψ ∈ H and bounded measurable function g on  we have    1 2   2  g dϕ | Eψ ≤ ϕ

|g| dψ | Eψ .   



Proof Let |ϕ Eψ| be the variation of the measure ϕ Eψ.3 It follows from the Radon-Nikodym theorem that there exists a measurable function u with values of modulus 1 such that ug ϕ Eψ = |g| |ϕ Eψ|.

 Sect. 4.2 we recalled the notion of total variation of a measure. Now the variation of a complex measure ν defined on a σ -algebra N is the map |ν| : N → [0 + ∞] defined by 3 In

|ν|() = sup

6 N

7   ν(n ) ,

∈N

n=1

with the supremum is taken over all measurable partitions of . Then |ν| is a measure and ν is absolutely continuous with respect to |ν|. Also 8  8    f dν  ≤ |f | d|ν|   for any integrable f .

10

138

Chapter 10 • Spectral Theorems

Therefore       g dϕ | Eψ ≤ |g| d|ϕ Eψ|   



=



    ug dϕ | Eψ = ϕ xug ψ =  ϕ xug ψ  ≤ ϕ

xug ψ ,



and since     

xug ψ 2 = ψ x|ug|2 ψ = ψ x|g|2 ψ = |g|2 dψ | Eψ, 

we find that    1 2   2  g dϕ | Eψ ≤ ϕ

|g| dψ | Eψ .   





10

Before going on to define integrals of not necessarily bounded functions with respect to spectral measures let us note that for bounded measurable functions f and g we have



      gf dϕ | Eψ = ϕ xf xg ψ = xf ϕ xg ψ = g dxf ϕ  Eψ,





which implies the following equality of complex measures: 

 xf ϕ Eψ = f ϕ Eψ .

(10.4)

Proposition 10.7 Let f be a measurable function  → C and define 6 7  Df = ψ ∈ H |f |2 dψ | Eψ < +∞ . 

Then Df is a dense subspace of H. Proof   For n ∈ N let n = ω ∈  |f (ω)| ≤ n . For ψ ∈ E(n )H and any measurable  ⊂  we have E()ψ = E()E(n )ψ = E( ∩ n )ψ, so that ψ Eψ () = ψ E()ψ = ψ E( ∩ n )ψ = ψ Eψ ( ∩ n ).

139 10.3 · Spectral Measure

Hence   |f |2 dψ | Eψ = |f |2 dψ | Eψ ≤ n2 ψ 2 < +∞ 

n

and thus E(n )H ⊂ Df for all n ∈ N. On the other hand, since χn −−−→ 1 pointwise, we have E(n )ψ −−−→ ψ. It follows that Df is a dense subset of H.

n→∞

n→∞

In order to see that Df is a vector subspace, take ψ, ϕ ∈ Df . Then for any  ∈ M  E()(ψ + ϕ) 2 ≤ E()ψ + E()ϕ 2 ≤ 2 E()ψ 2 + 2 E()ϕ 2 . In other words ψ + ϕ E(ψ + ϕ) ≤ 2 ψ Eψ + 2 ϕ Eϕ which shows that ψ +ϕ ∈ Df . The fact that Df is closed under scalar multiplication is clear. 

For a function f as in Proposition 10.7 we will now define an operator xf with domain D(xf ) = Df . Let us take ψ ∈ D(xf ) and ϕ ∈ H. By Lemma 10.6 and dominated convergence theorem, we have     12   2  f dϕ | Eψ ≤ ϕ

|f | dψ | Eψ ,   



and consequently  ϕ −→ f dϕ | Eψ 

is a bounded liner functional on H. It follows that there exists a unique vector η such that  f dϕ | Eψ = ϕ η . 

We put xf ψ = η. The left hand side of the above expression is linear in ψ, so xf is a densely defined linear operator. Moreover, for f bounded, this definition of xf coincides with the definition of xf given in Theorem 10.5 (see also  Sect. 4.3). ⓘ Remark 10.8 Let f1 and f2 be measurable functions  → C and let ψ ∈ D(xf1 ) ∩ D(xf2 ) = Df1 ∩ Df2 . Then for any ϕ ∈ H 

     ϕ xf1 ψ + ϕ xf2 ψ = f1 dϕ | Eψ + f2 dϕ | Eψ 

=







  (f1 + f2 ) dϕ | Eψ = ϕ xf1 +f2 ψ ,

10

140

Chapter 10 • Spectral Theorems

i.e. we have xf1 ψ + xf2 ψ = xf1 +f2 ψ.

Theorem 10.9 Let E be a spectral measure on (, M). Then (1) for any measurable function f :  → C the operator xf is closed, (2) for ψ ∈ D(xf ) we have

xf ψ 2 =



|f |2 dψ | Eψ,

(10.5)



(3) for any measurable f, g :  → C we have xf xg ⊂ xf g and, moreover, D(xf xg ) = Dg ∩ Df g , (4) for any measurable f :  → C we have xf ∗ = xf .

10

Proof We will prove (2), (3) and (4). Statement (1) follows by applying (4) to f .   Let χn be the characteristic function of ω ∈  |f (ω)| ≤ n and let fn = χn f . As fn is bounded, we have Df −fn = Df . Furthermore, by the dominated convergence theorem, for ψ ∈ Df we have  xf ψ − xf ψ 2 ≤ |f − fn | dψ | Eψ −−−→ 0 n n→∞



(10.6)

(cf. Remark 10.8 and the paragraph preceding it). Since fn is bounded, we can perform the following computation    xf ψ 2 = ψ x 2 ψ = |fn |2 dψ | Eψ, n |fn | 

and consequently (10.5) follows from the approximation (10.6) and (2) is proved. Assume for the moment that f is bounded. Then Dg ⊂ Df g and for ψ ∈ Dg , ϕ ∈ H we have 

       ϕ xf xg ψ = xf ϕ xg ψ = g dxf ϕ | Eψ = fg dϕ | Eψ = ϕ xf g ψ 



by (10.4). This shows that for ψ ∈ Dg (and f bounded) we have xf xg ψ = xf g ψ,

10

141 10.3 · Spectral Measure

and consequently  

|f |2 dxg ψ | Exg ψ = xf xg ψ 2 = xf g ψ 2  = |fg|2 dψ | Eψ,

 ψ ∈ D g(T ) .



Now since the above formula holds for all bounded measurable functions, it also holds for any measurable function f :  → C. Therefore xg ψ belongs to D(xf ) = Df if and only if ψ ∈ Df g , i.e. D(xf xg ) = Dg ∩ Df g . This proves Statement (3). For the proof of (4) we will once more use the approximation of f by the bounded functions fn . Take ψ ∈ Df and ϕ ∈ Df = Df . Then 

       ϕ xf ψ = lim ϕ xfn ψ = lim xfn ϕ ψ = xf ϕ ψ n→∞

n→∞

which shows that ϕ ∈ D(xf ∗ ) and xf ∗ ψ = xf ψ. In other words xf ⊂ xf ∗ .

(10.7)

Now take ζ ∈ D(xf ∗ ). By (3) we have xfn = xf xχn , so that xχn xf ∗ ⊂ (xf xχn )∗ = xfn ∗ = xfn , as xχn is a self-adjoint operator. In particular xχn xf ∗ ζ = xfn ζ,

n ∈ N.

We have 

|fn |2 dζ | Eζ  = xfn ζ 2 = xχn xf ∗ ζ 2



=



|χn |2 dxf ∗ ζ | Exf ∗ ζ 



  ≤ xf ∗ ζ Exf ∗ ζ () < +∞ and hence  

|f |2 dζ | Eζ  < +∞.

142

Chapter 10 • Spectral Theorems

It follows that ζ ∈ Df = D(xf ). This way we proved that D(xf ∗ ) ⊂ D(xf ) which together with (10.7) gives (4). 

Theorem 10.10 Let T be a self-adjoint operator. Then there exists a unique spectral measure ET on R such that T =



λ dET (λ). R

Before proving Theorem 10.10 let us note that the result allows us to define functional calculus for T for all (not necessarily bounded) Borel functions on R. More precisely for a Borel function f : R → C we put f (T ) =

 f (λ) dE(λ).

(10.8)

R

10

In case f is bounded, formula (10.8) produces the image of f under functional calculus for bounded Borel functions defined in Theorem 10.5. Moreover, it follows easily form Theorem 10.9(3) that if f and g are Borel functions R → C and g is bounded then f (T )g(T ) = (fg)(T ). Indeed: we have   D f (T )g(T ) = Dg ∩ Df g = H ∩ Df g = D (fg)(T ) , and therefore g(T )f (T ) ⊂ (gf )(T ) = (fg)(T ) = f (T )g(T ).

(10.9)

One cannot expect equality in (10.9), because  e.g. if f (T ) is unbounded and g = 0, then the domain of the left hand side is D f (T ) , while the right hand side is defined on all of H. Proof of Theorem 10.10 Let EzT be the spectral measure of the z-transform of T . Since σ (zT ) ⊂ [−1, 1] we can treat EzT as a Borel spectral measure on [−1, 1]. Moreover, we immediately see that since ±1 is  not an eigenvalue of zT , we have EzT {±1} = 0. In other words EzT is a spectral measure on ]−1, 1[. Let S=

1 −1

√μ

1−μ2

dEzT (μ).

10

143 10.3 · Spectral Measure

Let us check that S = T . Using - the notation introduced above, we have S = xf , where f (μ) = √ μ 2 . Put g(μ) = 1 − μ2 . Then 1−μ

Sxg ⊂ xf g =

1

μ dEzT (μ) = zT

−1

by Theorem 10.9(3). Moreover D(Sxg ) = Dg ∩ Df g = H, as g and fg are bounded on  = ]−1, 1[. This, in turn, means that Sxg = zT .

(10.10)

Finally note that 1

xg = g(zT ) = (1 − zT 2 ) 2 , which shows that (10.10) is in fact 1

1

S(1 − zT 2 ) 2 = T (1 − zT 2 ) 2 . In particular T ⊂ S, and since S and T are self-adjoint, we obtain S = T by Remark 8.9. Now let ET be the push-forward of EzT by the map ζ −1 : ]−1, 1[ → R, i.e.  ET (Δ) = EzT ζ (Δ) ,

 ∈ M,

where M is the σ -algebra of Borel subsets of R (in standard notation we write ET = (ζ −1 )∗ EzT ). Then 1

T =

√μ

1−μ2

−1

dEzT (μ) =

1 −1

ζ −1 (μ) dEzT (μ) =



λ dET (λ). R

We move on to the question of uniqueness of the spectral measure ET . Suppose that E is a Borel spectral measure on R such that T =



λ dE(λ). R

Then zT = ζ (T ) = the map ζ . Then zT =

1 −1

 R

μ dE (μ).

√λ

1+λ2

dE(λ). Now let E  be the push-forward of E onto ]−1, 1[ by

144

Chapter 10 • Spectral Theorems

It follows from the uniqueness of the spectral measure of zT that E  = EzT . Since ζ is a homeomorphism, we obtain E = (ζ −1 )∗ E  = (ζ −1 )∗ EzT , so that E = ET .



Let T be a self-adjoint operator and let f be a continuous real-valued function on R. In  Sect. 10.1. we defined f (T ) as the operator whose z-transform is (ζ ◦f )(T ) (having defined bounded continuous functions of T in Theorem 10.1). The obvious question is whether this operator is the same as f (T ) defined by formula (10.8). The following reasoning yields a positive answer to this question: let ET be the spectral measure of T and take a real-valued f ∈ C(R). Since



λ d(f∗ ET )(λ) =

R

10



f (λ) dET (λ) = f (T ),

R

it follows from uniqueness of the spectral measure of f (T ) that Ef (T ) = f∗ ET . Therefore   zf (T ) = ζ (μ) dEf (T ) (μ) = (ζ ◦ f )(λ) dET (λ) = (ζ ◦ f )(T ) R

R

where in the second equality we used a well-known property of a push-forward of measure. Consequently the z-transform of the operator f (T ) (defined by formula (10.8)) is the operator declared as the z-transform of f (T ) in Theorem 10.3.

Notes Spectral theory of unbounded operators has numerous applications and the reader will find far reaching extensions of the basic material of this chapter e.g. in [AkGl, Chapter VI], [Mau, Chapter VI], [ReSi1 , Chapter VIII] and in selected chapters of [Ped, Rud2]. All of these textbooks and monographs contain many examples and exercises on the subject.

11

145

Self-Adjoint Extensions of Symmetric Operators © Springer Nature Switzerland AG 2018 P. Sołtan, A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-319-92061-0_11

Self-adjointness of certain operators is crucial in many applications (e.g. in quantum physics or in partial differential equations). However, operators which naturally occur in many problems turn out to be symmetric, but not necessarily self-adjoint. It is for that reason that the problem of existence and classification of self-adjoint extensions of symmetric operators was one of the first challenges of the theory of unbounded operators on Hilbert spaces. In this chapter we will present the first results on that topic.

11.1 Containment of Operators in Terms of z-Transforms Theorem 11.1 Let S and T be densely defined closed operators on H. Then T ⊂ S if and only if 1

1

(1 − zS zS ∗ ) 2 zT = zS (1 − zT ∗ zT ) 2 .

Proof The relation T ⊂ S means exactly that G(T ) ⊂ G(S). Now the graphs of S and T have a precise description in terms of their z-transforms given in Theorem 9.6: 0 1 (1 − zT ∗ zT ) 2 ξ G(T ) = zT ξ

1 ξ ∈H ,

0 G(S) =

Moreover  G(T ) = UzT H ⊕ {0} ,

 G(S) = UzS H ⊕ {0} ,

1

(1 − zS ∗ zS ) 2 ξ zS ξ

1 ξ ∈H .

146

Chapter 11 • Self-Adjoint Extensions of Symmetric Operators

where  UzT =  UzS =

1

(1 − zT ∗ zT ) 2

−zT ∗

zT

(1 − zT zT ∗ ) 2

1

1

(1 − zS ∗ zS ) 2

−zS ∗

zS

(1 − zS zS ∗ ) 2

,

1

are unitary operators on H ⊕ H and H ⊕ {0} denotes the subspace of horizontal vectors 0 ξ 0

1 ξ ∈ H ⊂ H ⊕ H.

(cf. remarks following Theorem 9.9). It follows that G(T ) ⊂ G(S) is equivalent to   UzT H ⊕ {0} ⊂ UzS H ⊕ {0} .

(11.1)

Applying UzS ∗ to both sides of (11.1) we obtain  UzS ∗ UzT H ⊕ {0} ⊂ H ⊕ {0}.

11

It is easy to see that the only operators on H ⊕ H preserving the subspace H ⊕ {0} are the ones which in matrix notation have the form  •• . 0• In particular, the condition T ⊂ S is equivalent to vanishing of the lower left corner of the matrix UzS ∗ UzT  ∗ 1 1 (1 − zS ∗ zS ) 2 (1 − zT ∗ zT ) 2 −zS ∗ −zT ∗ = 1 1 ∗ zS (1 − zS zS ) 2 zT (1 − zT zT ∗ ) 2   1 1 (1 − zS ∗ zS ) 2 (1 − zT ∗ zT ) 2 zS ∗ −zT ∗ = 1 1 −zS (1 − zS zS ∗ ) 2 zT (1 − zT zT ∗ ) 2  • • = 1 1 ∗ ∗ −zS (1 − zT zT ) 2 + (1 − zS zS ) 2 zT • 1

1

In other words T ⊂ S if and only if (1 − zS zS ∗ ) 2 zT = zS (1 − zT ∗ zT ) 2 .



11

147 11.2 · Cayley Transform

11.2 Cayley Transform Before we continue, let us introduce a piece of notation which adds certain flexibility in dealing with partial isometries. Let v ∈ B(H) be a partial isometry. We know form Proposition 3.12 that there is a closed subspace S ⊂ H such that vξ = 0 for ξ ∈ S ⊥ and vξ = ξ for ξ ∈ S . We will denote by ˚ v the operator v S understood as an ∗ v ) = S = v H. In particular we can speak of the graph operator on H with domain D(˚ v as a subset of H ⊕ H. In accordance with our previous conventions for x ∈ B(H) of ˚ v + x is defined on D(˚ v + x) = D(˚ v ) = S . The operation v → ˚ v is a the operator ˚ bijection between the set of partial isometries on H and the set of isometric operators defined on closed subspaces of H. From now on let T be a symmetric operator. We are interested in self-adjoint extensions of T , i.e. operators S such that T ⊂ S and S = S ∗ . Since self-adjoint operators are closed and symmetric ones are closable (see Proposition 8.7), any selfadjoint extension of T is an extension of T . We can therefore assume that T is closed. It follows from Theorem 11.1 that (1 − zT ∗ zT ) 2 zT = zT ∗ (1 − zT ∗ zT ) 2 . 1

1

(11.2)

Define two operators w± = zT ± i(1 − zT ∗ zT ) 2 . 1

Then w+ and w− are isometries:  1  1 w± ∗ w± = zT ∗ ∓ i(1 − zT ∗ zT ) 2 zT ± i(1 − zT ∗ zT ) 2  1 1 = zT ∗ zT ∓ i (1 − zT ∗ zT ) 2 zT − zT ∗ (1 − zT ∗ zT ) 2 + (1 − zT ∗ zT ) = 1 by (11.2). Let W± = w± H and D± = W±⊥ . The subspaces D+ and D− are called the deficiency subspaces of T and their dimensions n± = dim D± are the deficiency indices of T . Proposition 11.2 We have D± = ker(T ∗ ∓ i1). Proof Since D± = W±⊥ , the condition ζ ∈ D± means that 

 1 ζ zT ξ ± i(1 − zT ∗ zT ) 2 ξ = 0,

ξ ∈ H.

(11.3)

Taking into account the description of the graph of T in terms of zT given in Theorem 9.6, we see that (11.3) is equivalent to ζ T ψ ± iψ = 0,

ψ ∈ D(T ),

148

Chapter 11 • Self-Adjoint Extensions of Symmetric Operators

 which means precisely that ζ ∈ D (T ± i1)∗ and (T ± i1)∗ ψ = 0. By Proposition 8.15(2) this amounts to ζ ∈ ker(T ∗ ∓ i1). 

Let cT = w− w+ ∗ . Then cT is a partial isometry with initial subspace W+ and final subspace W− . The operator c˚T is called the Cayley transform of the closed symmetric operator T . Using once again the argument from the proof of Proposition 11.2 we easily obtain 0 1 ϑ ϑ ∈ W+ G(c˚T ) = w− w+ ∗ ϑ 0 1 w+ ξ = ξ ∈H w− ξ 0 1  T ψ + iψ i1 1 = ψ ∈ D(T ) = G(T ) T ψ − iψ −i1 1 and hence

11

 −i1 i1 G(T ) = G(c˚T ). 1 1 In particular c˚T determines T . As G(T ) is a graph of a densely defined operator, by Propositions 8.3 and 8.1   G(T )⊥ ∩ H ⊕ {0} = G(T ) ∩ {0} ⊕ H = {0}. Therefore, for any ξ ∈ H we have    

ξ −i1 i1 ⊥ G(c˚T ) ⇒ ξ = 0 . 0 1 1

(11.4)

The condition on the left hand side of (11.4) means  ξ 0



−i1 i1 1 1



 ϑ c˚T ϑ

= 0,

ϑ ∈ D(c˚T )

which can be rewritten as 

 ξ (c˚T − 1)ϑ = 0,

ϑ ∈ W+ .

In other words, a vector which is orthogonal to (c˚T − 1)W+ = (cT − 1)cT ∗ H must be zero, i.e. the subspace (cT − 1)cT ∗ H is dense in H: (cT − 1)cT ∗ H = H.

(11.5)

149 11.2 · Cayley Transform

Notice further, that it follows from (11.5) noting T − 1) = {0}. Indeed:  that ker(c that for any partial isometry v we have ker v ∗ (v − 1) = ker v(v ∗ − 1) , we obtain ⊥    ker(cT − 1) ⊂ ker cT ∗ (cT − 1) = ker cT (cT ∗ − 1) = (1 − cT )cT ∗ H = {0}.1 Theorem 11.3 The mapping T → cT is a bijection between the set of closed symmetric operators on H and the set of partial isometries c ∈ B(H) satisfying (c − 1)c∗ H = H.

Proof We have already shown that if T is a closed symmetric operator then cT is a partial isometry satisfying (11.5) and that the map T → cT is injective (in other words cT determines T ). It remains to prove that if c is a partial isometry such that (c − 1)c∗ H = H then there exists a closed symmetric operator T such that c = cT . Let  −i1 i1 G= G(˚ c). 1 1 It is clear that G is a closed subspace of H ⊕ H. By reversing the reasoning leading up to condition (11.5), we obtain   

ξ ⊥ G ⇒ ξ = 0 . 0 More precisely, if     ξ −i1 i1 ϑ = 0, 0 1 1 ˚ cϑ

c), ϑ ∈ D(˚

then ϑ ∈ c∗ H,

ξ ⊥ (c − 1)ϑ,

i.e. ξ ⊥ (c − 1)c∗ H and this implies ξ = 0. Similarly we check that also the condition  

0 ∈G η

 ⇒



η=0



of the kernel of cT − 1 follows also from the fact that G(T ) does not contain non-zero vertical vectors, in a way similar to how we derived (11.5) from (11.4). It is not surprising that one of these conditions implies the other, because T is symmetric, i.e. 1 Triviality

G(T ) ⊂ G(T ∗ ) =

2

0 1 −1 0

3

G(T )⊥ .

11

150

Chapter 11 • Self-Adjoint Extensions of Symmetric Operators

holds. Indeed: if    ϑ −i1 i1 0 = 1 1 ˚ η cϑ for some ϑ ∈ D(˚ c) then η = (1 + ˚ c)ϑ and (˚ c − 1)ϑ = 0. Thus    ⊥ ϑ ∈ ker(˚ c−1) ⊂ ker(c−1) ⊂ ker c∗ (c−1) = ker c(c∗ −1) = (c−1)c∗ H = {0} c)ϑ = 0. (see remarks preceding the statement of the theorem) and consequently η = (1 + ˚ This way we have shown that G is a graph of a closed densely defined operator T (Corollary 8.4). Let us now look a the graph of T ∗ :   ⊥ −i1 i1 0 1 0 1 G(˚ G(T )⊥ = c) 1 1 −1 0 −1 0    ⊥   ⊥ 0 1 −i1 i1 1 1 = G(˚ c) = G(˚ c) . −1 0 1 1 i1 −i1 

G(T ∗ ) =

11

2 3 2 3 c) such that ξη = Now take an arbitrary ξη ∈ G(T ). Then there exists ϑ ∈ D(˚ 2 −i1 i1 32 ϑ 3  c) we have cϑ and for any ϑ ∈ D(˚ 1 1 ˚ 

−i1 i1 1 1

 ϑ ˚ cϑ



  1 1 ϑ i1 −i1 ˚ cϑ       i1 1 1 1 ϑ ϑ = ˚ −i1 1 i1 −i1 ˚ cϑ cϑ      ϑ 2i1 0 ϑ = ˚ 0 −2i1 ˚ cϑ cϑ      = 2i ϑ ϑ  − ˚ cϑ ˚ cϑ  = 0,

c preserves the scalar product on D(˚ c). This proves that as ˚ G(T ) ⊂ G(T ∗ ), i.e. T is symmetric.

ⓘ Remark 11.4 Let c1 and c2 be partial isometries on H such that c˚1 ⊂ c˚2 . Then if (c1 − 1)c1 ∗ H = H then also (c2 − 1)c2 ∗ H = H. Indeed: we have c2 ∗ H = D(c˚2 ) ⊃ D(c˚1 ) = c1 ∗ H and for any ξ ∈ D(c˚1 ) we have (c2 − 1)ξ = (c1 − 1)ξ . Therefore (c2 − 1)c2 ∗ H ⊃ (c1 − 1)c1 ∗ H.



11

151 11.2 · Cayley Transform

Corollary 11.5 Fix a closed symmetric operator T . Then T  is a closed symmetric extension of T if and only if c˚T ⊂ c˚ T .

Proof We already know that partial isometries c satisfying (c − 1)c∗ H = H are in bijection with closed symmetric operators and the operator corresponding to c is Tc defined by  G(Tc ) =

−i1 i1 G(˚ c). 1 1

(11.6)

Let s be a partial isometry such that ˚ c ⊂˚ s. Since (c − 1)c∗ H = H, Remark 11.4 shows ∗ that also (s − 1)s H = H. Furthermore, it is clear from formula (11.6) that the operator Ts defined by s satisfies G(Tc ) ⊂ G(Ts ) or, in other words, Tc ⊂ Ts . 

Theorem 11.6 Let T be a closed symmetric operator with deficiency subspaces D+ and D− . Let 0 ± = D

1 φ ±iφ

φ ∈ D± .

Then + ⊕ D − . G(T ∗ ) = G(T ) ⊕ D

(11.7)

Proof Let us first check that the three subspaces on the right hand side of (11.7) are pairwise orthogonal. It is clear that for any ξ, η ∈ H we have 



 ξ iξ

η −iη

= ξ η + iξ − iη = 0,

− are orthogonal. + and D and hence the subspaces D

152

Chapter 11 • Self-Adjoint Extensions of Symmetric Operators

Take ψ ∈ D(T ) and φ ∈ D± = ker(T ∗ ∓ i1). Then 



 ψ Tψ

φ ±iφ

= ψ φ ± i T ψ φ   = ψ φ ± i ψ T ∗ φ = ψ φ ± i ψ ± iφ = 0

− . + and to D which shows that G(T ) is orthogonal to D ± ⊂ G(T ∗ ) by definition of As T is symmetric, we have G(T ) ⊂ G(T ∗ ). Moreover D 2 ϕ 3 ∗ these subspaces (D± are eigenspaces of T ). It remains to show that if a vector T ∗ ϕ ∈ + ⊕ D − then ϕ = 0. Let us therefore take ϕ ∈ D(T ∗ ) such G(T ∗ ) is orthogonal to G(T ) ⊕ D that  ϕ − . + ⊕ D ⊥ G(T ) ⊕ D T ∗ϕ Then for any ψ ∈ D(T ) we have 

 ϕ

0=

11

T ∗ϕ

ψ Tψ



  = ϕ ψ + T ∗ ϕ T ψ ,

and so the functional   D(T )  ψ −→ T ∗ ϕ T ψ = − ϕ ψ is continuous. It follows that T ∗ ϕ ∈ D(T ∗ ) and T ∗ 2 ϕ = −ϕ, and hence  ϕ ∈ D (T ∗ + i1)(T ∗ − i1) and (T ∗ + i1)(T ∗ − i1)ϕ = 0.

(11.8)

Put η = (T ∗ − i1)ϕ, and note that it follows from (11.8) that η ∈ D− . Now for any vector η ∈ D− we have         i η η = i η (T ∗ − i1)ϕ = −iη T ∗ ϕ + i η − iϕ         ϕ η  ∗ = −iη T ϕ + η ϕ = = 0, T ∗ϕ −iη 2

η 3 −iη

− and we have assumed that ∈ D

2

ϕ 3 T ∗ϕ

− . Putting η = η we obtain ⊥ D 2 ϕ 3 + . η = 0, and consequently T ∗ ϕ = iϕ. This, in turn, means that ϕ ∈ D+ , i.e. T ∗ ϕ ∈ D 2 ϕ 3 + , so that ϕ = 0. However T ∗ ϕ is also orthogonal to D 

because

153 11.2 · Cayley Transform

From Theorem 11.6 we can immediately derive sufficient and necessary conditions for T to be self-adjoint. Corollary 11.7 Let T be a closed symmetric operator. Then the following conditions are equivalent: (1) T is self-adjoint, (2) D+ = D− = {0}, (3) n+ = n− = 0, (4) the Cayley transform of T is a unitary operator.

Similarly from Corollaries 11.7 and 11.5 we infer the following: Corollary 11.8 Let T be a closed symmetric operator. Then the following conditions are equivalent: (1) T has a self-adjoint extension, (2) there exists a unitary operator D+ → D− , (3) n+ = n− . Proof Self-adjoint extensions of T are in bijection with unitary extensions of c˚T and such an extension exists if and only if there is a unitary operator D+ → D− , i.e. if and only if n+ = n− . 

Note that Corollary 11.8 not only provides information on existence of self-adjoint extensions of a given symmetric operator T , but also on the number of such extensions. More precisely, there are as many extensions of T as there are unitary operators D+ → D− , i.e. infinitely many in any non-trivial case. In other words, self-adjoint extensions of T are parametrized by the unitary group U(n+ ). ⓘ Remark 11.9 A fully analogous analysis can be performed without the assumption that T is closed. In that case one must manage without the useful tool in the form of the z-transform, but the conclusion is practically the same as the results for closed symmetric operators given in Corollaries 11.7 and 11.8. The main difference lies in the fact that if T is not assumed to be closed, then it might happen that both deficiency indices are zero, but T is still not self-adjoint. In this case the closure of T is self-adjoint and we say that T is essentially self-adjoint.

We will end this section with a simple criterion for existence of self-adjoint extensions of a given symmetric operator. Let us begin by recalling that a mapping K : H → H is an anti-linear operator if it is additive: K(ξ + η) = K(ξ ) + K(η),

ξ, η ∈ H

11

154

Chapter 11 • Self-Adjoint Extensions of Symmetric Operators

and we have α ∈ C, ξ ∈ H.

K(αξ ) = αK(ξ ),

As for linear operators, we omit parentheses in the notation of an action of an anti-linear operator on a vector, i.e. we write Kξ instead of K(ξ ). An anti-linear operator J : H → H is called anti-unitary if J is surjective and ξ, η ∈ H

J ξ J η = η ξ  ,

(it is equivalent to J being an isometry of H onto H). Finally an anti-unitary operator is called an involution if J 2 = 1. Proposition 11.10 Let T be a symmetric operator and let J be an anti-unitary involution  on H. Assume that J D(T ) ⊂ D(T ) and T J ψ = J T ψ,

ψ ∈ D(T ).

Then T has a self adjoint extension.

11

Proof We have to show that the deficiency indices n± of T are equal. To that end we will check that J maps D+ bijectively onto D− . The operator J is R-linear, so dimR D+ = dimR D− and hence dim D+ = dim D− . Let ξ ∈ D± = ker(T ∗ ∓ i1). Then for any ψ ∈ D(T ) we have   J ξ T ψ = J ξ J 2 T ψ = J ξ J T J ψ = T J ψ ξ    = J ψ T ∗ ξ = J ψ ± iξ  = ±i J ψ ξ    = ±i J ψ J 2 ξ = ±i J ξ ψ = ∓iJ ξ ψ , and hence J ξ ∈ D(T ∗ ) and T ∗ J ξ = ∓iJ ξ . This shows that J D± ⊂ D∓ and it follows that D− = J 2 D− = J (J D− ) ⊂ J D+ ⊂ D− .

Consequently J D+ = D− .



11.3 Krein and Friedrichs Extensions Let T be a densely defined positive operator. Since for any ψ ∈ D(T ) we have ψ T ψ ≥ 0, it follows that ψ T ψ = T ψ ψ ,

ψ ∈ D(T ).

11

155 11.3 · Krein and Friedrichs Extensions

Hence, using the polarization formula, we obtain ψ T φ = T ψ φ ,

ψ, φ ∈ D(T ),

so that T is symmetric. This section will be devoted to studying self-adjoint extensions of positive operators. As we already mentioned in  Sect. 11.2, we can, without loss of generality, assume that T is closed. Now let T be positive and self-adjoint. Using spectral theory (or more precisely, functional calculus) one can show that σ (T ) ⊂ R+ . However, this fact can be derived using much more elementary methods (cf.  Sect. 9.1): (A) ker(T + 1) = {0}, because T ψ = −ψ for some ψ ∈ D(T ) implies 0 = ψ T ψ = − ψ 2 ≤ 0, i.e. ψ = 0. (B) The range of T + 1 is dense in H, as ξ ⊥ (T + 1)H means that ξ (T + 1)ψ ,

ψ ∈ D(T ),

i.e. D(T )  ψ → ξ T ψ is a continuous linear functional and consequently ξ ∈ D(T ∗ ) = D(T ) and T ξ = T ∗ ξ = −ξ . In other words ξ ∈ ker(T + 1) = {0}. (C) For any ψ ∈ D(T ) we have (T + 1)ψ 2 = T ψ 2 + ψ T ψ + T ψ ψ + ψ 2 ≥ ψ 2 .

(11.9)

 In particular the range of T + 1 is closed: if (T + 1)ψn n∈N is a Cauchy sequence then it follows from (11.9) that also (ψn )n∈N is a Cauchy sequence. Let φ = lim ψn . Then from closedness of T +1 we infer that lim (T +1)ψn = (T +1)φ. n→∞

n→∞

Summing up the above observations we see that T + 1 is a bijection D(T ) onto H and the inverse operator (T +1)−1 is continuous (in fact it is a contraction). In particular 1 −1 ∈ σ (T  1 ). Furthermore, for any λ > 0 the operator λ T is positive and self-adjoint, so −1 ∈ σ λ T . It follows that −λ ∈ σ (T ). This way we have shown that σ (T ) ⊂ R+ . Lemma 11.11 Let K be a closed subspace of the Hilbert space H and let a ∈ B(K), b ∈ B(K, K⊥ ) and c ∈ B(K⊥ ). Then  a b∗ ≥0 (11.10) b c

156

Chapter 11 • Self-Adjoint Extensions of Symmetric Operators

in B(H) if and only if a ≥ 0 in B(K) and for any ε > 0 we have c ≥ b(a + ε1K )−1 b∗

(11.11)

in B(K⊥ ). Proof 2 3 2 a b∗ 32 ξ 3 we easily check that for (11.10) to hold it Computing the expectation value ξ0 b c 0 is necessary that a ≥ 0 in B(K). Let pK ∈ B(H) be the projection onto K. Then for any ε > 0 the operator εpK is positive and εpK −−→ 0 in strong topology (monotonically decreasing). It is clear that ε$0

(11.10) is equivalent to  a + ε1K b∗ ≥ 0, b c

ε > 0.

(11.12)

Assume (11.12) holds. Then for any ε > 0 we have     a + ε1K b∗ ξ 1 ξ1 ≥ 0, ξ2 ξ2 b c

11

for arbitrary

2 ξ1 3 ξ2

∈ K ⊕ K⊥ . In other words

  ξ1 (a + ε1K )ξ1  + ξ1 b∗ ξ2 + ξ2 bξ1  + ξ2 cξ2  ≥ 0,

ξ1 ∈ K , ξ2 ∈ K ⊥ .

Setting ξ1 = −(a + ε1K )−1 b∗ ξ2 we obtain 

 ξ2 − b(a + ε1K )−1 b∗ ξ2 + ξ2 cξ2  ≥ 0,

ξ2 ∈ K ⊥

which is exactly (11.11). Now assume that a ≥ 0 and for any ε > 0 the relation (11.11) holds. We have 

  a + ε1K b∗ 0 0 a + ε1K b∗ + = 0 c − b(a + ε1K )−1 b∗ b c b b(a + ε1K )−1 b∗

and the element  0 0 0 c − b(a + ε1K )−1 b∗

11

157 11.3 · Krein and Friedrichs Extensions

is obviously positive. Therefore, in order to prove (11.12), it is enough to show that for any ε > 0 we have  a + ε1K b∗ ≥ 0. b b(a + ε1K )−1 b∗ Writing A for a + ε1K we get  ξ1 ξ2

   a + ε1K ξ1 b∗ b b(a + ε1K )−1 b∗ ξ2

    = ξ1 Aξ1  + ξ1 b∗ ξ2 + ξ2 bξ1  + ξ2 bA−1 b∗ ξ2 .

Assume for the moment that ξ2 = bη + η with η ∈ ker b∗ = (bK)⊥ . Then  ξ1 ξ2

   a + ε1K ξ1 b∗ b b(a + ε1K )−1 b∗ ξ2   = ξ1 Aξ1  + ξ1 b∗ bη + b∗ η     + bη + η bξ1 + bη + η bA−1 b∗ (bη + η )   = ξ1 Aξ1  + ξ1 b∗ bη + b∗ η

    + b∗ bη + b∗ η ξ1 + b∗ bη + b∗ η A−1 (b∗ bη + b∗ η )       = ξ1 Aξ1  + ξ1 b∗ bη + b∗ bη ξ1 + b∗ bη A−1 b∗ bη     = ξ1 Aξ1 + b∗ bη + b∗ bη ξ1 + A−1 b∗ bη     = ξ1 A(ξ1 + A−1 b∗ bη) + b∗ bη ξ1 + A−1 b∗ bη     = Aξ1 ξ1 + A−1 b∗ bη + b∗ bη ξ1 + A−1 b∗ bη   = Aξ1 + b∗ bη ξ1 + A−1 b∗ bη   = A(ξ1 + A−1 b∗ bη) ξ1 + A−1 b∗ bη   = ξ1 + A−1 b∗ bη A(ξ1 + A−1 b∗ bη) ≥ 0.

(11.13)

158

Chapter 11 • Self-Adjoint Extensions of Symmetric Operators

Now any vector ξ2 is a limit of vectors of the form (11.13), because K⊥ is the direct sum K⊥ = bK ⊕ ker b∗ . This way we have shown that 

 ξ1 ξ2 for all

2 ξ1 3 ξ2

a + ε1K b∗ b b(a + ε1K )−1 b∗



 ξ1 ξ2

≥0

∈ K ⊕ K⊥ .



Corollary 11.12 Let H, K, a, b, c be as in Lemma 11.11. Then  a b∗ 0≤ ≤1 b c

(11.14)

if and only is 0 ≤ a ≤ 1K and for any ε > 0 we have b(a + ε1K )−1 b∗ ≤ c ≤ 1K⊥ − b(1K − a + ε1K )−1 b∗ .

11

Proof By Lemma 11.11 the first inequality of (11.14) is equivalent to a ≥ 0 and b(a +ε1K )−1 b∗ ≤ c for any ε > 0. The second inequality of (11.14) is, in turn equivalent to  1K − a −b∗ 0≤ −b 1K⊥ − c

(11.15)

and applying Lemma 11.11 to this case we find that (11.15) is equivalent to a ≤ 1K and 1K⊥ − c ≥ b(1K − a + ε 1K )−1 b∗ for any ε > 0. The latter condition can be expressed also as c ≤ 1K⊥ − b(1K − a + ε 1K )−1 b∗ for any ε > 0. This way we proved that (11.14) is equivalent to the condition that 0 ≤ a ≤ 1K and b(a + ε1K )−1 b∗ ≤ c ≤ 1K⊥ − b(1K − a + ε 1K )−1 b∗ ,

ε, ε > 0,

which is further equivalent to b(a + ε1K )−1 b∗ ≤ c ≤ 1K⊥ − b(1K − a + ε1K )−1 b∗ ,

ε > 0. 

Now let T and S be positive and self-adjoint operators. We write T ≥ S if the bounded operators (T + 1)−1 and (S + 1)−1 satisfy (T + 1)−1 ≤ (S + 1)−1 (cf. Proposition 3.7)

11

159 11.3 · Krein and Friedrichs Extensions

Theorem 11.13 Let T be a closed positive operator. Then there exist positive self-adjoint operators TK and TF such that (1) T ⊂ TK and T ⊂ TF , ≤ (2) a positive self-adjoint operator T is an extension of T if and only if TK ≤ T TF .

Proof Let K be the range of the operator T + 1. The arguments used to prove Statements (A) and (C) preceding Lemma 11.11, show that K is a closed subspace of H and T + 1 is a bijection of D(T ) onto K not decreasing the norm. It follows that the linear map (T + 1)−1 : K −→ D(T ) is continuous. Let pK and pK⊥ be the projections H → K and H → K⊥ and define a = pK (T + 1)−1 ∈ B(K),

b = pK⊥ (T + 1)−1 ∈ B(K, K⊥ ).

Take an arbitrary ζ ∈ K and let (T + 1)−1 ζ = ξ . Writing both sides in the decomposition H = K ⊕ K⊥ we have   ξ1 aζ = , bζ ξ2 since in this notation ζ =

2ζ 3 0

 ζ aζ  = ζ ξ1  =

, we have 

ζ 0

 ξ1 0



 =

ζ 0

 ξ1 ξ2

= ζ ξ  = (T + 1)ξ ξ  = ξ (T + 1)ξ  = ξ 2 + ξ T ξ  ≥ ξ 2 = aζ 2 + bζ 2 , because T is symmetric. In other words, we obtain a ≥ a ∗ a + b∗ b = a 2 + b∗ b which can also be written as a(1K − a) ≥ b∗ b.

(11.16)

 + 1), and Now let T be a positive and self-adjoint extension of T . Then (T + 1) ⊂ (T it easily follows that  (T + 1)−1 K = (T + 1)−1 .

160

Chapter 11 • Self-Adjoint Extensions of Symmetric Operators

In particular the matrix of the operator (T + 1)−1 corresponding to the decomposition H = K ⊕ K⊥ must be of the form  (T + 1)−1 =

a b∗ b c

(11.17)

(b∗ in the upper right corner is a consequence of self-adjointness of (T + 1)−1 ). Moreover, since (T + 1)−1 is a positive contraction, we have  a b∗ 0≤ ≤ 1, b c so by Corollary 11.12 b(a + ε1K )−1 b∗ ≤ c ≤ 1K⊥ − b(1K − a + ε1K )−1 b∗

(11.18)

for all ε > 0. In particular, for T to have a positive self-adjoint extension T it is necessary that b(a + ε1K )−1 b∗ ≤ 1K⊥ − b(1K − a + ε1K )−1 b∗ ,

11

ε > 0.

(11.19)

Fix ε > 0. The inequality in (11.19) is equivalent to  b (a + ε1K )−1 + (1K − a + ε1K )−1 b∗ ≤ 1K⊥ . Also, we have (a + ε1K )−1 + (1K − a + ε1K )−1 = f (a), where f (t) = all t ∈ [0, 1]. Moreover, since f (t) =

(11.20) 1 t+ε

+

1 1−t+ε

for

1+2ε (t+ε)(1−t+ε) ,

we can rewrite (11.20) as  (1 + 2ε)b (a + ε1K )−1 (1K − a + ε1K )−1 b∗ ≤ 1K⊥ 1

1

or, putting d = (a + ε1K )− 2 (1K − a + ε1K )− 2 , as (1 + 2ε)bd ∗ db∗ ≤ 1K⊥ .

(11.21)

To prove that (11.21) holds, we note first that (11.16) is simply F (a) ≥ b∗ b, where F (t) = t (1 − t) is a continuous function on [0, 1] of norm 14 . Therefore b∗ b ≤ 14 1K (see Proposition 3.5) and consequently 1K ≥ 4b∗ b ≥ 2b∗ b.

(11.22)

11

161 11.3 · Krein and Friedrichs Extensions

Now, using (11.22), we compute (a + ε1K )(1K − a + ε1K ) = a(1K − a) + ε(1K − a) + ε2 1K ≥ a(1K − a) + (ε + ε2 )1K ≥ b∗ b + (ε + ε2 )1K ≥ b∗ b + (ε + ε2 )2b∗ b = (1 + 2ε)b∗ b + ε2 b∗ b ≥ (1 + 2ε)b∗ b, which can then be successively rewritten as d −1 d ∗ −1 ≥ (1 + 2ε)b∗ b and then as 1K ≥ (1 + 2ε)db∗ bd ∗ . This, in turn, means that (1 + 2ε)db∗ bd ∗ ≤ 1. Thus 1 ≥ (1 + 2ε) bd ∗ 2 = (1 + 2ε) db∗ 2 = (1 + 2ε) bd ∗ db∗

which implies (11.21), and we have proved (11.19). Now we notice that as ε $ 0, the left hand side of (11.19) is monotonically increasing, while the right hand side is monotonically decreasing. Thus, by Theorem 3.15, the left hand side has a supremum in and he right hand side an infimum in B(K⊥ ) which are the limits of either side in strong topology as ε $ 0. Define cF = sup b(a + ε1K )−1 b∗ , ε>0

cK = inf 1K⊥ − b(1K − a + ε1K )−1 b∗ . ε>0

Then c ∈ B(K⊥ ) satisfies (11.18) for all ε > 0 if and only if cF ≤ c ≤ cK .

(11.23)

We will now show that the mapping T → c is an order reversing bijection between the set of positive self-adjoint extensions of T and the set of c ∈ B(K⊥ ) satisfying (11.23). We already know that any positive self-adjoint extension T of T defines c satisfying (11.23) via (11.17). Furthermore, if T and T are positive self-adjoint extensions of T and c and c are the corresponding operators on K⊥ then

T ≥ T



(T + 1)−1 ≤ (T + 1)−1    ∗  ab a b∗ ⇐⇒ ≤ b c b c   

0 0 ⇐⇒ ≥ 0 ⇐⇒ c ≥ c .  0 c −c ⇐⇒



162

Chapter 11 • Self-Adjoint Extensions of Symmetric Operators

Finally, if c ∈ B(K⊥ ) satisfies (11.23) then  a b∗ 0≤ ≤1 b c (by Corollary 11.12). Also, the range of 

2a

b∗ b c

3

is dense in H, because

  a b∗ a b∗ a b∗  ⊥ H= K ⊕ {0} (K ⊕ K ) ⊃ b c b c b c 0 1 aζ = ζ ∈ K = (T + 1)−1 H = D(T ). bζ

2 ∗3 2 ∗ 3−1 Hence ker ab bc = {0}, and consequently ab bc is a closed densely defined operator. Moreover, it is self-adjoint and positive. In fact it satisfies 

11

a b∗ b c

−1

≥ 1.

In particular  a b∗  T = b c

−1

−1

is positive and self-adjoint. Clearly T is an extension of T . Let TK and TF be the positive self-adjoint extensions of T corresponding to c = cK and c = cF respectively: (TK + 1)

−1

 a b∗ = , b cK

(TF + 1)

−1

 a b∗ = b cF

Then any positive self-adjoint extension T of T must satisfy TK ≤ T ≤ TF and any positive self-adjoint operator T satisfying (11.24) is an extension of T .

(11.24) 

Theorem 11.13 shows that any positive operator has self-adjoint extensions and that among those which are positive there is a minimal and a maximal one (in the sense of the partial order on positive operators). The extensions TK and TF are called respectively the Krein extension and the Friedrichs extension of the positive operator T .

163 11.3 · Krein and Friedrichs Extensions

Notes The reader will find further information on self-adjoint extensions of symmetric operators in [AkGl, Chapter VII], [Mau, Chapter V], [ReSi1 , Chapter VIII], [ReSi2 , Chapter X]. The Krein and Friedrichs extensions are usually presented in the context of quadratic forms on Hilbert spaces. This approach is taken e.g. in [Kat, Chapter 6], [ReSi2 , Chapter X]. The version of the theory presented above is less popular, but requires fewer preliminary results and can be carried out using exclusively bounded operators.

11

12

165

One-Parameter Groups of Unitary Operators © Springer Nature Switzerland AG 2018 P. Sołtan, A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-319-92061-0_12

One of particularly fruitful applications of the theory of operators on Hilbert spaces is in representation theory of topological groups. In this chapter we will study basic properties of representation theory of the abelian group R. A family (ut )t∈R of operators on H is called a strongly continuous one-parameter group of unitary operators if (1) for any t ∈ R the operator ut is unitary, (2) for all t, s ∈ R we have ut+s = ut us , (3) for any vector ψ ∈ H the map R  t −→ ut ψ ∈ H

(12.1)

is continuous. In view of (2) condition (3) is equivalent to the fact that for any ψ ∈ H the mapping (12.1) is continuous at 0 ∈ R. Consider the following example: let H be a self-adjoint operator on H and put ut = exp(−itH ). It is easy to see that conditions (1) and (2) are satisfied. Condition (3) follows from continuity of Borel functional calculus, because as t → 0, the functions λ → exp(−itλ) converge pointwise to the constant function 1 (and are all uniformly bounded). It follows that (ut )t∈R is a strongly continuous one-parameter group of unitary operators. In order to shorten our notation we will from now on write e−itH instead of exp(−itH ).

166

Chapter 12 • One-Parameter Groups of Unitary Operators

12.1 Stone’s Theorem

Theorem 12.1 Let H be a self-adjoint operator on H and for t ∈ R let ut = e−itH . Then (1) if ψ ∈ D(H ) then lim ti (ut ψ − ψ) = H ψ, t→0

(2) if the limit lim ti (ut ϕ − ϕ) exists then ϕ ∈ D(H ). t→0

Proof 1 Ad (1). Take ψ ∈ D(H ). Then ψ = (1 + H 2 )− 2 ξ for some ξ ∈ H and we have i t (ut ψ

− ψ) = Ft (H )ξ,

where Ft (λ) =

1 i e−itλ −1 λ(1 + λ2 )− 2 . t λ

The functions Ft are uniformly bounded and they converge pointwise to ζ (defined at the beginning of  Sect. 10.1), as t → 0. Thus lim i (ut ψ t→0 t

12

− ψ) = zH ξ = H ψ.

 putting Ad (2). Let us define an operator H   ) = ϕ ∈ H the limit lim i (ut ϕ − ϕ) exists D(H t t→0

) and for ϕ ∈ D(H ϕ = lim i (ut ϕ − ϕ). H t t→0

 is a linear operator. Moreover H ⊂ H , so H  is densely defined. It is clear that H ) the calculation Furthermore, for ϕ1 , ϕ2 ∈ D(H 

   ϕ2 = lim ϕ1 i (ut ϕ2 − ϕ2 ) ϕ1 H t t→0

= lim

t→0



   ϕ1 ti ut ϕ2 − ϕ1 ti ϕ2

= lim

  −i

   ut ∗ ϕ1 ϕ2 − −i t ϕ1 ϕ2

= lim

  −i

   u−t ϕ1 ϕ2 − −i t ϕ1 ϕ2

t→0

t→0

= lim

t

t



i (u−t ϕ1 t→0 −t

   ϕ1 ϕ2 − ϕ1 ) ϕ2 = H

12

167 12.1 · Stone’s Theorem

 is symmetric. By Remark 8.9 we have H  = H. shows that H



We can draw the following conclusion from Theorem 12.1: let H be a self-adjoint operator and let ψ0 ∈ D(H ). Then the initial value problem = H , i d dt

(0) = ψ0

(12.2)

(with unknown function  : R → H) has a global solution. Indeed: defining (t) = e−itH ψ0 we obtain a continuous function  : R → H such that ▬ (0) = ψ0 , ▬  is differentiable at 0,  = H ψ0 . ▬ i d dt t=0 Moreover, for any t ∈ R we have1 1 s

  (t + s) − (t) = e−itH 1s (s) − (0) −−→ e−itH H ψ0 s→0

= H e−itH ψ0 = H (t) which means that  is differentiable everywhere and is the solution of the initial value problem (12.2). Furthermore, the solution of the problem (12.2) is unique. Indeed: if  : R → H is 2 a solution then putting f (t) = (t) − (t) we obtain a differentiable scalar-valued function such that f (0) = 0 and d f (t) dt

=

d dt

(t) − (t) (t) − (t)

= −iH (t) − iH (t) (t) − (t) + (t) − (t) − iH (t) − iH (t) = 0. Thus (t) = (t) for all t. A similar reasoning is used in the prof of the next theorem. Theorem 12.2 (Stone’s Theorem) Let (ut )t∈R be a strongly continuous one-parameter group of unitary operators on H. Then there exits a self-adjoint operator H on H such that ut = e−itH .

of (10.9) to f (λ) = λ and g(λ) = e−itλ yields e−itH H ⊂ H e−itH . Moreover, using  Theorem 12.1(2) one can easily show that for any s we have e−isH D(H ) ⊂ D(H ). It follows that  −itH itH −itH −itH D He = e D(H ) = D(H ), and consequently e H = He . 1 Application

168

Chapter 12 • One-Parameter Groups of Unitary Operators

Proof For ϕ ∈ H and f ∈ C∞ c (R) (smooth functions with compact support) define ϕf =



f (t)ut ϕ dt R

  and let D = span ϕf ϕ ∈ H, f ∈ C∞ c (R) . Now if (fn )n∈N is a sequence of smooth functions of compact support such that for any n 3 2 ▬ supp f ⊂ − n1 , n1 ,   n ▬ fn (t) dt = 1 R

then for any ϕ ∈ H

ϕfn

  − ϕ = fn (t)ut ϕ dt − fn (t)ϕ dt R

R

 f (t)(u ϕ − ϕ) dt = n t R

  ≤ fn (t) ut ϕ − ϕ dt ≤ sup ut ϕ − ϕ −−−→ 0. n→∞

|t|≤ n1

R

It follows that D is a dense subspace of H. Note that if f ∈ C∞ c (R) and we denote by fs the function t → f (t − s) then

12

us ϕf = us



f (t)ut ϕ dt =

R



f (t)us ut ϕ dt R

=



f (t)us+t ϕ dt =

R



f (t − s)ut ϕ dt = ϕfs

R

which shows that the operators us preserve the subspace D . Furthermore, for ϕf ∈ D we have    1 1 (u ϕ − ϕ ) = f (t − s)u ϕ dt − f (t)u ϕ dt f t t s s f s =

 R

because

fs −f s

R

R

f (t−s)−f (t) ut ϕ dt −−→ s s→0



− f  (t) ut ϕ dt = −ϕf  ,

R

−−→ −f  uniformly. s→0

Define a linear operator H0 on H putting D(H0 ) = D and H0 φ = i lim 1s (us φ − φ), s→0

φ ∈ D.

Notice that the range of H0 is contained in D and for any t ∈ R we have H0 ut = ut H0 .

12

169 12.1 · Stone’s Theorem

Now in the same way as in the proof of Theorem 12.1(2) we show that the operator H0 is symmetric: for φ, ψ ∈ D we have ψ H0 φ = lim

i s→0 s

ψ us φ − φ = lim

−i s→0 −s

= lim

i s→0 s

ψ (us − 1)φ

(u−s − 1)ψ φ = −i iH0 ψ φ = H0 ψ φ .

We will now show that H0 is essentially self-adjoint, i.e. H0 is self-adjoint. For this we need to prove that ker(H0 ∗ ± i1) = {0} (see Corollary 11.7). Take η ∈ ker(H0 ∗ ± i1). Then for any φ ∈ D(H0 ) we have d dt

 η ut φ = lim η s→0

1 s (ut+s

− ut )φ



  = lim η ut 1s (us − 1)φ

(12.3)

s→0

= η ut (−i)H0 φ = −i η H0 ut φ   = −i H0 ∗ η ut φ = −i ∓iη ut φ = ± η ut ψ . This means that the scalar-valued function g : t → η ut φ satisfies g  = ±g, so that g(t) = g(0)e±t ,

t ∈ R.

  On the other hand g(t) ≤ η

ut φ = η

φ , so g must be constant and hence equal to 0. In particular η φ = g(0) = 0. Since this holds for all φ ∈ D and D is dense in H, we obtain η = 0. Put H = H0 . Then H is a self-adjoint operator and we can consider the one-parameter  group e−itH t∈R . Take φ ∈ D and let ξ(t) = ut φ − e−itH φ,

t ∈ R.

Then for any t we have ξ(t) ∈ D ⊂ D(H ) and a computation analogous to (12.3) shows that d dt ξ(t)

= −iH0 ut φ + iH e−itH φ = −iH ξ(t),

t ∈ R.

Consequently



d 2 dt ξ(t)

=

d dt

ξ(t) ξ(t) =

d

dt ξ(t)

  ξ(t) + ξ(t)

d dt ξ(t)



= −iH ξ(t) ξ(t) + ξ(t) − iH ξ(t) = 0

170

Chapter 12 • One-Parameter Groups of Unitary Operators

and hence the function t → ξ(t) is constant. On the other hand ξ(0) = 0, so we get ξ(t) = 0 for all t, i.e. ut φ = e−itH φ,

t ∈ R, φ ∈ D .

Now the density of D in H implies that ut = e−itH for all t ∈ R.



Let (ut )t∈R be a strongly continuous one-parameter group of unitary operators on H. The self-adjoint operator H such that ut = e−itH for all t is obviously unique and we call it the infinitesimal generator of the group (ut )t∈R .

12.2 Trotter Formula Let H be a self-adjoint operator. Recall that D(H ) is a Hilbert space with the graph norm:

ψ H =

ψ 2 + H ψ 2 ,

ψ ∈ D(H ).

We also know that for any ξ ∈ H the function R  t → e−itH ξ ∈ H is continuous and if ψ ∈ D(H ) then for any t e−itH ψ ∈ D(H )

12

and the function R  t → e−itH ψ ∈ D(H ) is continuous for the norm · H . Indeed: it is enough to show continuity at zero (cf.  Sect. 12.1) and this, in light of the fact that H e−itH = e−itH H , follows from the following computation 2 2  −itH 2 e ψ−ψ H = e−itH ψ − ψ + H e−itH ψ − ψ 2 2 = e−itH ψ − ψ + e−itH H ψ − H ψ −−→ 0. t→0

Theorem 12.3 (Trotter Formula) Let H and K be self-adjoint operators and assume that H + K is self-adjoint. Then for any t ∈ R  −i t H −i t K n e n e n −−−→ e−it (H +K) n→∞

in strong topology.

171 12.2 · Trotter Formula

Proof For t = 0 let F (t) =

1 t

 −itH −itK e e − e−it (H +K) .

Now for any ξ ∈ H the function t → F (t)ξ is continuous on R K {0}, because the maps 1 : t −→ 1t e−it (H +K) ξ

and

2 : t −→ 1t e−itK ξ

are continuous and we have 1 −itH −itK e ξ te

= e−itH 2 (t).

Therefore for t, t  ∈ R K {0} 1 −itH −itK   e e ξ − 1t e−it H e−it K ξ t  = e−itH 2 (t) − e−it H 2 (t  )  ≤ e−itH 2 (t) − e−itH 2 (t  ) + e−itH 2 (t  ) − e−it H 2 (t  )  0. = 2 (t) − 2 (t  ) + e−i(t−t )H 2 (t  ) − 2 (t  ) −−→  t→t

It is moreover clear that lim F (t)ξ = 0. t→±∞

Let us denote the domain D(H + K) = D(H ) ∩ D(K) by D. We will check that for ψ ∈ D we also have lim F (t)ψ = 0.

t→0

Indeed: we have  −itH −itK  e e ψ − ψ − 1t e−it (H +K) ψ − ψ    = e−itH 1t e−itK ψ − ψ + 1t e−itH ψ − ψ − 1t e−it (H +K) ψ − ψ

F (t)ψ =

1 t

and since 1 t→0 t

lim

lim 1 t→0 t lim 1 t→0 t



e−it (H +K) ψ − ψ = −i(H + K)ψ,



e−itH ψ − ψ = −iH ψ,



e−itK ψ − ψ = −iKψ,

12

172

Chapter 12 • One-Parameter Groups of Unitary Operators

we find that  lim F (t)ψ = −iH ψ − iKψ − − i(H + K) ψ = 0.

t→0

 In particular for ψ ∈ D w can put F (0)ψ = 0 and this way we obtain a family F (t) t∈R of linear maps D → H such that the function t → F (t)ψ is continuous. Moreover lim F (t)ψ = lim F (t)ψ = 0.

t→0

t→±∞

(12.4)

Consider on D the norm · H +K in which this space is a Hilbert space. Since · H +K ≥

· , we see that all operators F (t) are continuous from D (with the norm · H +K ) to H. It follows from continuity of t → F (t)ψ and (12.4) that for any ψ ∈ D the set   F (t)ψ t ∈ R is bounded. there exists a constant M such Therefore, by the Banach-Steinhaus theorem, that F (t) ≤ M for all t ∈ R, where by F (t) we mean the norm of an operator  D, · H +K → H. In other words, for any ψ ∈ D and t ∈ R we have F (t)ψ ≤ M ψ H +K . Fix a ψ ∈ D and let

12

  C = e−is(H +K) ψ s ∈ [−1, 1] . As we explained before formulating the theorem, the function s −→ e−is(H +K) ψ   is continuous R → D, · H +K and hence C is compact in D, · H +K . ε Take now ε > 0 and let φ1 , . . . , φN be an 2M -net in C. For any η ∈ C there exists i such ε that η − φi H +K < 2M . Furthermore, there exists δ > 0 such that for |t| < δ we have F (t)φj < ε , 2

j = 1, . . . , N,

and therefore F (t)η ≤ F (t)η − F (t)φi + F (t)φi ≤ M η − φi H +K + F (t)φi < ε  which shows that the functions ψ → F (t)ψ t∈[−1,1] converge to zero uniformly on C, as t → 0. In other words, for a fixed ψ ∈ D the quantity F (t)e−is(H +K) ψ

12

173 12.2 · Trotter Formula

tends to 0, as t → 0, uniformly in s ∈ [−1, 1]. Now the identity  −i t H −i t K n e n e n ψ − e−it (H +K) ψ

 t n  t n t = e−i n H e−i n K − e−i n (H +K) ψ =

n−1   −i t H −i t K n  −i t H −i t K  t n−1−m t e n e n e n e n − e−i n (H +K) e−i n (H +K) ψ m=0

implies that  −i t H −i t K n e n e n ψ − e−it (H +K) ψ  t t (n−1−m) t t ≤ n max e−i n H e−i n K − e−i n (H +K) e−i n (H +K) ψ 0≤m≤n−1

= |t|

max

0≤m≤n−1

 t −i t (n−1−m) (H +K) F n e ψ n

 ≤ |t| max F nt e−is(H +K) ψ −−−→ 0 |s| 0 such that for any T ∈ T we have T ≤ M.

Proof For x ∈ X let us denote   M(x) = sup T x T ∈ T .   Suppose that sup T T ∈ T = +∞. Then there exists a sequence (Tn )n∈N of elements of T and a sequence (yn )n∈N of unit vectors in X such that

Tn yn > 4n ,

n ∈ N.

Putting xn = 21n yn we obtain a sequence (xn )n∈N converging to zero and such that for all n we have Tn xn > 2n . We will now choose a strictly increasing sequence of natural numbers (nk )k∈N so that for all k ∈ N ⎧ k

⎪ ⎪ ⎨ T x

>1+k+ M(x ), nk+1 nk+1

⎪ ⎪ ⎩ xn < k+1

1 2k+1



j =1

nj

 −1  . max Tnj j = 1, . . . , k

176

APPENDIX A

We put n1 = 1 and once n1 , . . . , np have been chosen, we use the facts that xn −−−→ 0 and

Tn xn −−−→ +∞ to choose np+1 in such a way that np+1 > np and

n→∞

n→∞

⎧ ⎪ ⎪ ⎨ Tn

p

xnp+1 > 1 + p + M(xnj ), j =1

 −1 ⎪  ⎪ 1 ⎩ xnp+1 < p+1 . max Tnj j = 1, . . . , p 2 p+1

In particular, since Tn yn > 4n , we have Tn ≥ 4n for all n, and hence   max Tnj j = 1, . . . , k > 1. Consequently for any j

xnj <

1 2j

It follows that



 −1  < max Tns s = 1, . . . , j − 1



j =1



xnj < +∞, so that the series

j =1

1 2j

.

xnj converges to some x ∈ X.

Now we have   ∞  ∞ k

Tnk+1 x = T x = T x + T x + T x nk+1 nj nk+1 nj nk+1 nk+1 nk+1 nj j =1

j =1

j =k+2

   k ∞  = T x − − T x − T x n n n n n n k+1 k+1 k+1 j k+1 j j =1

j =k+2

 ∞  k ≥ Tnk+1 xnk+1 − T x + T x nk+1 nj nk+1 nj j =1

j =k+2

k k ∞    >1+k+ M(xj ) − Tnk+1 xnj + Tnk+1 xnj j =1

≥1+k+

k  j =1

≥1+k+

k  j =1

j =1

j =k+2

∞ k   − M(xj ) − T x T x n n n n k+1 j k+1 j j =1

M(xj ) −

k 

j =k+2

∞ 

Tnk+1 xnj −

j =1

The last sum on the right hand side is smaller than

Tnk+1

xnj .

j =k+2 ∞

j =k+2

1 2j

< 1, because for j ≥ k + 2 we

have

xnj <

1 2j



 −1  ≤ max Tnj j = 1, . . . , j − 1

1 1 . 2j Tnk+1

177 APPENDIX A

k

Tnk+1 xnj is majorized by M(xj ). This shows that Tnk+1 x > j =1 j =1   k, which contradicts boundedness of the set T x T ∈ T and consequently shows that   sup T T ∈ T < +∞. 

Furthermore the sum

A.2

k

Dynkin’s Theorem

Let  be a set. A family P of subsets of  is called a π -system if it is closed under finite intersections. Furthermore, a family L of subsets of  is a λ-system if the following conditions are satisfied: 1.  ∈ L, 2. if  ∈ L then  ∈ L, ∞  3. if (n )n∈N is a sequence of pairwise disjoint elements of L then n ∈ L. n=1

Theorem A.2 (Dynkin’s Theorem on π- and λ-Systems) Let P be a π-system and L a λ-system of subsets of some set . Assume that P ⊂ L. Then the σ -algebra generated by P is contained in L.

ⓘ Remark A.3 Before proving Theorem A.2 let us note that a λ-system is a σ -algebra if and only if it is also a π-system. Indeed: on one hand, if L is a σ -algebra, then it is a π-system. On the other hand, if L is a λ-system and a π-system at the same time then for any sequence (n )n∈N of elements of L the sets  n = n K 

 n−1 9

 k

= n ∩ (1 ∩ · · · ∩ n−1 )

k=1

 n )n∈N are pairwise disjoint and also belong to L. Moreover ( ∞ 9 n=1

n =

∞ 9

 n ∈ L. 

n=1

This shows that L is closed under arbitrary countable unions of its elements, i.e. it is a σ -algebra. Proof of Theorem A.2 Denote by λ(P ) the smallest λ-system containing the family P and by σ (P ) the smallest σ -algebra containing P . Clearly λ(P ) ⊂ σ (P ). We will show that λ(P ) = σ (P ), and hence we will have σ (P ) ⊂ L, because λ(P ) ⊂ L.

(A.1)

178

APPENDIX A

To prove (A.1) one needs to show that λ(P ) is a σ -algebra. In view of Remark A.3 we have see that λ(P ) is a π -system (is closed under intersections). Fix  ∈ λ(P ) and let   L =  ⊂   ∩  ∈ λ(P ) . We will check that L is a λ-system: (1) we have  ∩  =  ∈ λ(P ), and so  ∈ L , (2) if  ∈ L then   ∩  =  ∪  ∩  = ( ∩ ) ∩    = ( ∩ ) ∪  ∈ λ(P ), because  ∩  and  are disjoint elements of λ(P ), (3) if (n )n∈N is a sequence of pairwise disjoint elements of L then 9 ∞

 ∞ 9 n ∩  = (n ∩ ) ∈ λ(P ),

n=1

n=1

as the latter is a countable union of disjoint elements of λ(P ). Now since P is closed under finite intersections, for any  ∈ P we have P ⊂ L , and as L is a λ-system, we also have λ(P ) ⊂ L . In other words, for  ∈ λ(P ) and  ∈ P we have  ∩  ∈ λ(P ). This, in turn, means that P ⊂ L for any  ∈ λ(P ) and consequently λ(P ) ⊂ L .

(A.2)

Now we notice that (A.2) says that λ(P ) is a family closed under finite intersections.

A.3



Tensor Product of Hilbert Spaces

Let H and K be Hilbert spaces. Their tensor product as vector spaces will be denoted by H ⊗alg K. We will prove that there exists a scalar product on H ⊗alg K such that ψ1 ⊗ ϕ1 ψ2 ⊗ ϕ2  = ψ1 ψ2  ϕ1 ϕ2  ,

ψ1 , ψ2 ∈ H, ϕ1 , ϕ2 ∈ K.

Fix ψ1 ∈ H and ϕ1 ∈ K and consider the mapping

H × K  (ψ2 , ϕ2 ) −→ ψ1 ψ2  ϕ1 ϕ2  .

179 APPENDIX A

It is a bi-linear functional, and so it determines a unique linear functional ψ1 ,ϕ1 on H ⊗alg K. Let E be the space of anti-linear functionals on H ⊗alg K. Then the map

H × K  (ψ1 , ϕ1 ) −→ ψ1 ,ϕ1 ∈ E is bi-linear and consequently it defines a linear  : H ⊗alg K → E such that 

(ψ1 ⊗ ϕ1 ) (ψ2 ⊗ ϕ2 ) = ψ1 ,ϕ1 (ψ2 ⊗ ϕ2 ) = ψ1 ψ2  ϕ1 ϕ2  = ψ2 ψ1  ϕ2 ϕ1  .

Now for ξ, η ∈ H ⊗alg K let us put ξ η = (ξ )(η) = (ξ )(η).

(A.3)

It is clear that · · is a sesquilinear from. Moreover, taking ξ = ψ1 ⊗ϕ1 and η = ψ2 ⊗ϕ2 we obtain ψ1 , ψ2 ∈ H, ϕ1 , ϕ2 ∈ K.

ψ1 ⊗ ϕ1 ψ2 ⊗ ϕ2  = ψ1 ψ2  ϕ1 ϕ2  , Now if ξ=

N 

ξi1 ⊗ ξi2 ,

η=

M 

ηj1 ⊗ ηj2 ,

j =1

i=1

then     N M 1 2 1 2 ξ η = (ξ )(η) =  ξi ⊗ ξi ηj ⊗ ηj j =1

i=1

=

M N  

(ξi1 ⊗ ξi2 )(ηj1 ⊗ ηj2 )

i=1 j =1

=

M  N     ηj1 ξi1 ηj2 ξi2 i=1 j =1

=

M N   

  ξi1 ηj1 ξi2 ηj2 .

i=1 j =1

This immediately shows that η ξ  = ξ η,

ξ, η ∈ H ⊗alg K.

180

APPENDIX A

Furthermore, for ξ=

N 

ξi1 ⊗ ξi2

i=1

we can always assume that the vectors ξ12 , . . . , ξN2 are orthonormal (by applying the Gram-Schmidt orthogonalization process), so if ξ = 0 then ξ ξ  =

N 

ξ1i 2 > 0.

i=1

This means that (A.3) defines a scalar product on H ⊗alg K. Moreover, it is clear that the condition ψ1 ⊗ ϕ1 ψ2 ⊗ ϕ2  = ψ1 ψ2  ϕ1 ϕ2  ,

ψ1 , ψ2 ∈ H, ϕ1 , ϕ2 ∈ K

determines this scalar product uniquely. This way H ⊗alg K becomes a normed space and we will denote by H ⊗ K its completion. Clearly H ⊗ K is a Hilbert space and we call it the tensor product of the Hilbert spaces H and K. Let {ψi }i∈I and {ϕj }j ∈J be orthonormal bases in H and K respectively. Then the system {ψi ⊗ ϕj }i∈I, j ∈J is orthonormal and it spans a dense subspace of H ⊗ K. It follows that it is an orthonormal basis of H ⊗ K. Let (, μ) and (, ν) be σ -finite measure spaces such that the Hilbert spaces H = L2 (, μ) and K = L2 (, ν) are separable. Choose orthonormal bases {ψi }i∈N and {ϕj }j ∈N of H and K. Then the system of functions  ×   (ω, λ) −→ ψi (ω)ϕj (λ),

i, j ∈ N

(A.4)

is orthonormal in L2 ( × , μ ⊗ ν). We will show that it forms an orthonormal basis of this space. Take f ∈ L2 ( × , μ ⊗ ν) and suppose that



f (ω, λ)ψi (ω)ϕj (λ) dμ(ω)dν(λ) = 0

×

for all i and j . Notice that since |f |2 is integrable over  × , the function f (ω, ·) belongs to L2 (, ν) for almost all λ. Therefore we can write 0=

 



  f (ω, λ)ψi (ω) dμ(ω) ϕj (λ) dν(λ) = f (·, λ) ψi  ϕj (λ) dν(λ). 

181 APPENDIX A

Since this holds for all j , the function λ → f (·, λ) ψi  is equal to zero almost everywhere, ∞ i.e. on  K i for some i of measure zero. This means that for λ ∈  K i the function f (·, λ) is zero almost everywhere on . Consequently f i=1

is equal tozero almost everywhere on  × . Since ψi ⊗ ϕj }i,j ∈N is an orthonormal basis of H ⊗ K and (A.4) is an orthonormal basis of L2 ( × , μ ⊗ ν), we can define an operator u : L2 (, μ) ⊗ L2 (, ν) −→ L2 ( × , μ ⊗ ν) mapping each ψi ⊗ ϕj to the function  ×   (ω, λ) −→ ψi (ω)ϕj (λ) ∈ C. As u maps an orthonormal basis onto an orthonormal basis, it is unitary. This proves the following: Proposition A.4 Let (, μ) and (, ν) be σ -finite measure spaces such that the Hilbert spaces H = L2 (, μ) and K = L2 (, ν) are separable. Then the operator L2 (, μ) ⊗ L2 (, ν) −→ L2 ( × , μ ⊗ ν) mapping ψ ⊗ ϕ to the function  ×   (ω, λ) −→ ψ(ω)ϕ(λ) ∈ C is unitary.

A.4

Open Mapping Theorem

In the formulation and proofs of all statements below we will use the convention of Banach space theory according to which the closed unit ball in a Banach space X is denoted by X1 . Also the closed ball with center 0 and radius r in X will be denoted by Xr . Theorem A.5 (Open Mapping Theorem) Let X and Y be Banach spaces and let T ∈ B(X, Y) be surjective. Then T is an open mapping.

182

APPENDIX A

Proof We will show that T (X1 ) contains a neighborhood of 0 ∈ Y. For any n ∈ N define a new norm · n on Y setting  

y n = inf u X + n v Y u ∈ X, v ∈ Y, y = v + T u . Furthermore let   Z = f : N → Y sup f (n) n < +∞ . n∈N

With the norm

f Z = sup f (n) n ,

f ∈Z

n∈N

the vector space Z becomes a Banach space.1 Define a sequence (Sn )n∈N of bounded operators from Y to Z: ⎧ ⎨y, (Sn y)(m) = ⎩0,

m = n, m = n.

We have Sn y Z = y n for all y ∈ Y. Take y ∈ Y and n ∈ N. Since y = y + T 0, we see that

y n ≤ n y Y , while by surjectivity of T we can choose x ∈ X such that y = 0 + T x, and hence

y n ≤ x X . The latter estimate does not depend on n, so for any y the set {Sn y n ∈ N} is bounded in Z. Thus, by the Banach-Steinhaus theorem, there exists a constant M > 0 such that Sn ≤ M for all n. 3 12 Now fix δ ∈ 0, M . We will show that T (X1 ) contains the open ball   BY (δ) = y ∈ Y y Y < δ . Let y ∈ BY (δ). Then for any n we have

y n = Sn y Z ≤ M y Y < Mδ < 1.

1 However,

completeness of Z will not be relevant.

183 APPENDIX A

It follows that for any n there are un ∈ X and vn ∈ Y such that y = vn + T un and

un X + n vn Y < 1. In particular vn Y <

1 n,

and consequently T un −−−→ y. Moreover, for any n we have n→∞

un X < 1, so y ∈ T (X1 ).  We now proceed to show that it follows from BY (δ) ⊂ T (X1 ) that BY 2δ ⊂ T (X1 ). Let   y ∈ BY 2δ . We know that y ∈ T X 1 , so there exists x1 ∈ X 1 such that y − T x1 Y < 4δ . 2  δ 2 , there exists x2 ∈ X 1 such that (y − T x1 ) − T x2 < Furthermore since y − T x1 ∈ BY 4

4

δ 8.

Y

Continuing this procedure we obtain a sequence (xn )n∈N such that for any n we have

xn X < 21n and n  y − T xk < Y

k=1

It follows that y =



n ∈ N.

δ , 2n+1

 T xk = T

k=1



 xk .

k=1

This way we proved that for any x0 ∈ X the image of a closed ball with center x0 under T contains an open ball with center T x0 . It now easily follows that T is an open map. 

Corollary A.6 Let T ∈ B(X, Y) be a bijection. Then the inverse of T is bounded. Proof The operator T is an open map, because it is surjective. It follows that T −1 is continuous.  Corollary A.7 (Closed Graph Theorem) Let T : X → Y be a linear map such that the graph of T , i.e. the set 1

0 G(T ) =

x Tx

x∈X

is closed in X ⊕ Y. Then T is bounded. Proof 2 3 The closed subspace G(T ) ⊂ X ⊕ Y is a Banach space (e.g. with the norm Txx =   max x , T x , cf.  Sect. 8.1). Let PX and PY be the maps  x → x ∈ X, − Tx  x PY : G(T )  → T x ∈ X. − Tx PX : G(T ) 

184

APPENDIX A

PX and PY are obviously continuous. Moreover PX is a bijection, and hence an invertible  operator: PX−1 ∈ B X, G(T ) . Therefore T = PY PX−1 

is continuous.

A.5

Quotient Spaces and Algebras

A.5.1 Quotients of Banach Spaces Let X be a Banach space and let S be a closed subspace of X. The so called quotient norm on the quotient vector space X/S is defined as

x + S X/S = inf x + u X ,

x ∈ X.

u∈S

Let us check that this expression does, in fact, define a norm: ▬ obviously we have α(x + S) X/S = |α| x + S X/S , ▬ for x, y ∈ X (x + S) + (y + S) = x + y + S X/S X/S = inf x + y + u X = inf x + y + u + v X u∈S

u,v∈S

 ≤ inf x + u X + y + v X u,v∈S

=





inf x + u X + inf y + v X

u∈S

v∈S

= x + S X/S + y + S X/S , ▬ if x + S X/S = 0 then for any n there exists un ∈ S such that x − (−un ) = x + un X ≤ 1 , n X so that it follows from closedness of S that x ∈ S, i.e. x + S = 0. Furthermore X/S with the norm defined above is a Banach space. To prove this let us first notice that the quotient map q : X −→ X/S is obviously a contraction. Now let (xn + S)n∈N be a Cauchy sequence in X/S. Choose a subsequence (xnk )k∈N of (xn )n∈N such that (xn + S) − (xn + S) < k k+1 X/S

1 . 2k

(A.5)

185 APPENDIX A

Then let us define a sequence (uk )k∈N of elements of S as follows: we put u1 = 0 and choose u2 such that

xn1 − xn2 + u2 X ≤ xn1 − xn2 + S X/S + 12 . Next we select u3 such that (xn + u2 ) − (xn + u3 ) ≤ xn − xn + S X/S + 2 3 2 3 X

1 22

etc. The procedure yields (uk )k∈N such that (xn + uk ) − (xn + uk+1 ) ≤ xn − xn + S X/S + k k+1 k k+1 X

1 , 2k

k ∈ N.

Now thanks to the estimate (A.5), we have (xn + uk ) − (xn + uk+1 ) < k k+1 X

1 , 2k−1

k ∈ N,

 so that (xnk + uk )k∈N is a Cauchy sequence in X (as the sequence of norms xnk + uk X k∈N is summable). Let x0 be its limit. Then clearly xnk + S = q(xnk + uk ) −−−→ k→∞

q(x0) = x0 +S, because the map q is continuous. But (xn +S)n∈N is a Cauchy sequence, so once we found it to have a convergent subsequence (with limit x0 + S), it also must converge: xn + S −−−→ x0 + S. n→∞

In particular X/S is a Banach space.

Theorem A.8 Let X and Y be Banach spaces and T ∈ B(X, Y). Put S = ker T . Then the canonical map T : X/S → Y making the diagram T

X

BB BB BB BB q

/ Y > } }} } }} }} T

X/S commutative is bounded with T = T .

186

APPENDIX A

Proof Take x ∈ X and u ∈ S. Then

T x Y = T (x + u) Y ≤ T

x + u X , and hence

T x Y ≤ T inf x + u X = T

x + S X/S . u∈S

Of course T(x + S) = T(qx) = T x, so that T ≤ T . On the other hand, for any ε > 0 there exists xε ∈ X such that xε X = 1 and T x Y > T − ε. Thus we have  T (xε + S) Y = T xε Y > T − ε, and also xε + S X/S ≤ xε X = 1. It follows that T ≥ T .



Let us note that the closed graph theorem can be used to show that if the range of T in Theorem A.8 is additionally assumed to be closed then T is an isomorphism of X/S onto T X (see Corollaries A.6 and A.7).

A.5.2 Ideals and Quotients of C∗ -Algebras Let A be a C∗ -algebra. A left ideal in A is a subspace L ⊂ A such that

y ∈ L, x ∈ A



⇒



xy ∈ L .

Similarly a subspace R ⊂ A is a right ideal if

y ∈ R, x ∈ A



⇒



yx ∈ R .

It is easy to see that L is a left ideal if and only if the set   L∗ = y ∗ y ∈ L is a right ideal. An ideal is two-sided if it is both a left ideal and a right one.2

2 Generally,

the terms “ideal” and “two-sided ideal” are used as synonyms.

187 APPENDIX A

Theorem A.9 Let A be a C∗ -algebra with unit3 and let L ⊂ A be a left ideal (not necessarily closed). Then there exists a net (ei )i∈I of elements of L such that (1) for any i ∈ I we have 0 ≤ ei ≤ 1, (2) i  j implies ei ≤ ej , (3) for any a ∈ L we have a − aei −−→ 0. i∈I

Proof Denote by I the set of pairs (n, F ) such that n ∈ N and F is a finite subset of L. We introduce a partial order on I by declaring that (n, F )  (n , F  ) whenever n ≤ n and F ⊂ F  . For i = (n, F ) ∈ I we put vi =



b∗ b,

ei =

1

n 1 + vi

−1

vi .

b∈F

In other words ei = fn (vi ), where fn (t) =

and since 0 ≤ fn ≤ 1 for all n, we have 0 ≤ ei ≤ 1.

t 1 n +t

Take i, j ∈ I such that i = (n, F )  j = (n , F  ). Then vi ≤ vj , so that 1 n

+ vi

−1

1



n

+ vj

−1

(A.6)

by Proposition 4.25. On the other hand, since for all t ∈ R+ we have 

1 1 n n

+t

−1





1 1 n n

+t

−1

,

also the inequality 

1 1 n n

+ vj

−1





1 1 n n

+ vj

−1

(A.7)

must hold. Putting together (A.6) and (A.7) we obtain 

1 1 n n

3 The

+ vi

−1





1 1 n n

+ vj

−1





1 1 n n

+ vj

−1

,

assumption of A being unital is not crucial for this theorem (nor for any of the remaining results of this section), but it makes the proof less cumbersome and Statement (1) easier to express.

188

APPENDIX A

and hence 1−



1 1 n n

+ vi

−1

≤ 1−



1 1 n n

+ vj

−1

(A.8)

.

But ei =

1

n 1 + vi

−1

vi =

1

n1

+ vi

−1  1

n 1 + vi

 −1 − n1 1 = 1 − n1 n1 + vi

(A.9)

 −1 and similarly ej = 1 − n1 n1 + vj , so (A.8) means that ei ≤ ej . Finally let us take a ∈ L and consider i = (n, F ) such that a ∈ F . Thanks to (A.9) we have  ∗  b(1 − ei ) b(1 − ei ) = (1 − ei )vi (1 − ei ) =



1 1 n2 n

+ vi

−2

vi ≤

1 4n 1,

b∈F

because

1 n

+t

−2

t≤

n 4

for all t ∈ R+ . Consequently

 ∗   ∗  a(1 − ei ) a(1 − ei ) ≤ b(1 − ei ) b(1 − ei ) ≤

1 4n 1,

b∈F

and this gives  ∗ 

a − aei 2 = a(1 − ei ) a(1 − ei ) ≤

1 4n .

It follows that a − aei −−−→ 0. n→∞



The net (ei )i∈I constructed in Theorem A.9 is an example of an approximate unit for the left ideal L. Corollary A.10 Let J be a closed ideal in a unital C∗ -algebra A. Then J is self-adjoint. Proof Let (ei )i∈I be an approximate unit for J. Then for any a ∈ J we have a = lim aei . i∈I

Now, since the ideal J is two sided, for any i we have ei a ∗ ∈ J and consequently a ∗ = lim ei a ∗ i∈I

belongs to the closed subset J.



Proposition A.11 Let J be a closed ideal in a unital C∗ -algebra A. Then the quotient space A/J with quotient norm is a C∗ -algebra.

189 APPENDIX A

Proof Since J is a self-adjoint subset (and a two-sided ideal), it is clear that the quotient space is a unital ∗-algebra. Furthermore, as J is a closed subspace, the quotient space A/J is a Banach space with the quotient norm

a + J = inf a + u ,

a ∈ A.

u∈J

Moreover, the quotient norm satisfies (a + J)(b + J) ≤ a + J

b + J ,

a, b ∈ A

and

a ∗ + J = a + J ,

a ∈ A.

The latter of these properties is obvious, while the former is a consequence of the fact that if ε > 0 and u, v ∈ J are such that

a + u ≤ a + J + ε

b + v ≤ b + J + ε

and

then

ab + J ≤ ab + av + ub + uv = (a + u)(b + v)  ≤ a + u

b + v ≤ a + J

b + J + ε a + J + b + J + ε . It remains to show that for any a ∈ A we have the inequality a + J 2 ≤ a ∗ a + J . Let (ei )i∈I be an approximate unit for J. It is clear that

a + J ≤ a − aei ,

i ∈ I,

because aei ∈ J, and hence

a + J ≤ inf a − aei . i∈I

On the other hand, for any u ∈ J we have

a + u ≥ (a + u)(1 − ei ) , since 1 − ei ≤ 1. Therefore

a + u ≥ lim inf (a + u)(1 − ei ) = lim inf (a − aei ) + (u − uei ) i∈I

i∈I

= lim inf (a − aei ) ≥ inf (a − aei ) , i∈I

i∈I

190

APPENDIX A

where in the first equality we used the fact that u − uei −−→ 0. As a result we get i∈I

a + J = inf a − aei , i∈I

a ∈ A.

Using this we compute: 2

a + J 2 = inf a(1 − ei ) = inf (1 − ei )a ∗ a(1 − ei ) i∈I

i∈I

≤ inf a ∗ a(1 − ei ) = a ∗ a + J . i∈I



The first part of the next theorem is a simple generalization of Proposition 4.9 from  Sect. 4.2. Theorem A.12 Let A and B be unital C∗ -algebras and let  : A −→ B be a unital ∗-homomorphism. Then  is a contraction and (A) is a unital C∗ subalgebra4 of B. If  is injective then it is isometric.

Proof Clearly for any a ∈ A the spectrum of the element (a) in B is contained in the spectrum of algebra A. Now a is self-adjoint then a is equal to |σ (a)| and (a) equals  a in the if  σ (a) . Therefore (a) ≤ a . Now for an arbitrary b ∈ A, putting a = b∗ b, we obtain (b) 2 = (b∗ b) = (a) ≤ a = b∗ b = b 2 and consequently  is a contraction. Assume now that  is injective. In light of the reasoning above, to see that  is an isometry, it is enough to prove that it preserves norms of self-adjoint elements. This, in turn,  will follow once we establish that for a = a ∗ we have σ (a) = σ (a). We already know   that σ (a) ⊂ σ (a), so suppose that σ (a)  σ (a). Then there exists a continuous  function f on σ (a) such that f = 0, but f = 0 on σ (a) . We have   f (a) =  f (a) ,

4 Closed

∗-subalgebra.

191 APPENDIX A

 because this holds for polynomials and  is continuous. However, since f = 0 on σ (a) we find that f (a) ∈ ker  = {0}. In other words f (a) = 0 which in view of Theorem 4.23  means that f = 0. The resulting contradiction shows that σ (a) = σ (a). Coming back to the case of possibly non-injective  we will prove that the range of  is closed. Indeed: we can express  as the composition of the quotient map A → A/ ker  and  of A/ ker  onto (A) ⊂ B. The map   is an injective unital the canonical isomorphism  ∗-homomorphism from the C∗ -algebra A/ ker  to the C∗ -algebra B, so it is isometric and the range of an isometry is closed. 

ⓘ Remark A.13 An important consequence of the fact that injective ∗-homomorphisms

are isometric is the uniqueness of the norm on a C∗ -algebra (formally speaking we only proved this for unital algebras). This follows by noting that the identity map must then be isometric for any two C∗ -norms on a given C∗ -algebra.

Let us consider now a special case, where the arbitrary unital C∗ -algebra A is replaced by the algebra C(X) of continuous functions on a compact topological space X. Let J be a closed ideal in C(X). Furthermore let   Y = x ∈ X f (x) = 0 for all f ∈ J . Then Y is a closed subset of X, since Y =

:

f −1 (0).

f ∈J

Note that the set X K Y is in a natural way a locally compact topological space. Indeed: any compact space is a Tikhonov  space, so for any x0 ∈ X K Y there exists a continuous function f on X such that f Y = 0 and f (x0 ) = 1. Then for any compact neighborhood N of x0 in X the set   x ∈ X f (x) ≥ 12 ∩ N is a compact neighborhood of x0 in X K Y . Denote by C0 (X K Y ) the algebra whose elements are all continuous functions on X vanishing on the subset Y . With norm inherited from C(X) the algebra C0 (X K Y ) becomes a C∗ -algebra (without unit). Moreover, it is easy to see that C0 (X K Y ) is naturally isomorphic to the algebra of continuous functions on the  topological space XK Y which vanish at infinity, i.e. such f that for any δ > 0 the set x ∈ X K Y |f (x)| ≥ δ is compact in X K Y . Clearly J ⊂ C0 (X K Y ). Furthermore ▬ for any x ∈ X K Y there exists f ∈ J such that f (x) = 0, ▬ for any x1 , x2 ∈ X K Y there exists f ∈ J such that f (x1 ) = f (x2 ) (indeed: there exists f ∈ J such that f(x1 ) = 0 and there is g ∈ C(X) such that g(x1 ) = 1, g(x2 ) = 0; it follows that f = fg satisfies f ∈ J and f (x1 ) = g(x1 ) = 0 = f (x2 )).

192

APPENDIX A

Therefore it follows from the Stone-Weierstrass theorem for locally compact spaces that J = C0 (X K Y ). Theorem A.14 Denote J = C0 (X K Y ). Then the map  C(X)/J  f + J −→ f Y ∈ C(Y ) is an isometric ∗-isomorphism C(X)/C0 (X K Y ) onto C(Y ).

Proof  The map C(X)/J  f + J → f Y is a unital ∗-homomorphism which is one-to-one (by definition) and surjective (by Tietze’s theorem). Isomorphisms of C∗ -algebras are isometric. 

193

Index of Notation H H1 H⊕K H ⊕ {0} {0} ⊕ H H ⊗alg K H⊗K span S spanS · · · ·Tr ψ ψ Tr B(H) B(H, K) B(H)+ Proj B(H) B0 ( H ) B1 ( H ) B2 ( H ) F (H ) 1 C(X) C0 (XKY ) Cb (X) Cc (X) (R) C∞ c  · ∞ B (X) C[ · ] X1 (Xr )  · X/S H  (D) L2 (, μ) L∞ (, μ)  · 2  · 1

Hilbert space closed unit ball in H direct sum of Hilbert spaces space of horizontal vectors space of vertical vectors algebraic tensor product tensor product of Hilbert spaces linear hull of a set S closed linear hull of a set S scalar product scalar product on the space of Hilbert-Schmidt operators “bra” operator defined by vector ψ “ket” operator defined by vector ψ trace space of bounded operators on H space of bounded operators H → K set of positive operators projections acting on a Hilbert space H space of compact operators on H space of trace class operators on H space of Hilbert-Schmidt operators on H set of finite dimensional operators on H identity operator, unit of a C∗ -algebra space of continuous functions on X space of continuous functions on XKY vanishing at infinity space of bounded continuous functions on X space of continuous functions on X with compact support space of smooth functions on R with compact support uniform norm space of bounded Borel functions on X space of polynomials with complex coefficients closed unit ball of radius 1 (radius r) in a Banach space X quotient norm space of functions holomorphic on a neighborhood of a set D integral over a closed oriented curve space of square-integrable functions on  space of essentially bounded functions on  norm in L2 space, Hilbert-Schmidt norm trace norm

© Springer Nature Switzerland AG 2018 P. Sołtan, A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-319-92061-0

194

Index of Notation

Mf Vess (f ) f= ˙g ˙ f ≤g χ  μ⊗ν C∗ (x) C∗ (x, 1) σ (x) |σ (x)| ρ(x) σ (x1 , . . . , xn ) l(x) r(x) s(x) Re x, Im x |x| 1 x2 x+ , x − λ dEx (λ)

operator of multiplication by f essential range of f equality almost everywhere inequality almost everywhere characteristic function of  complement of  tensor product of measures (product measure) C∗ -algebra generated by x C∗ -algebra generated by x and 1 spectrum of x spectral radius of x resolvent set of x joint spectrum of x1 , . . . , xn left support of x right support of x support of x real and imaginary parts of x modulus of x square root of x positive and negative parts of x expression of x as a spectral integral

σ (x)

xf ξ Eξ  ξ Eη D(T ) G(T ) T  · T T∗ ζ zT T ⊂S ˚ v c˚T D± n±

integral of f with respect to a spectral measure positive measure associated to spectral measure E complex measure associated to spectral measure E domain of T graph of T closure of T graph norm adjoint operator t the function t → √1+t 2 z-transform of T containment of operators isometric operator defined by partial isometry v Cayley transform of T deficiency subspaces deficiency indices

195

References [AkGl] N.I. Akhiezer, I.M. Glazman, Theory of Linear Operators in Hilbert Space (Dover Publications, Mineola, 1993) [Arv1 ] W. Arveson, An Invitation to C∗ -Algebras (Springer, New York, 1976) [Arv2 ] W. Arveson, A Short Course of Spectral Theory (Springer, New York, 2002) [Eng] R. Engelking, General Topology (Heldermann Verlag, Berlin, 1989) [Hal] P. Halmos, A Hilbert Space Problem Book (Springer, New York, 1982) [Kat] T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin, 1980) [Lan] E.C. Lance, Hilbert C∗ -Modules. A Toolkit for Operator Algebraists (Cambridge University Press, Cambridge, 1995) [Mau] K. Maurin, Methods of Hilbert Spaces (Polish Scientific Publishers, Warsaw, 1972) [Ped] G.K. Pedersen, Analysis Now (Springer, New York, 1995) [ReSi1 ] M. Reed, B. Simon, Methods of Modern Mathematical Physics I. Functional Analysis (Academic Press, London, 1980) [ReSi2 ] M. Reed, B. Simon, Methods of Modern Mathematical Physics II. Fourier Analysis, Self-Adjointness (Academic Press, London, 1980) [Rud1 ] W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1987) [Rud2 ] W. Rudin, Functional Analysis (McGraw-Hill, New York, 1991) [WoNa] S.L. Woronowicz, K. Napiórkowski, Operator theory in the C∗ -algebra framework. Rep. Math. Phys. 31, 353–371 (1991) ˙ [Zel] W. Zelazko, Banach Algebras (Polish Scientific Publishers, Warsaw, 1973)

© Springer Nature Switzerland AG 2018 P. Sołtan, A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-319-92061-0

197

Index absolute value of an operator, 28, 126 adjoint of an unbounded operator, 105 anti-linear operator, 153 anti-unitary operator, 154 approximate unit, 188 Banach algebra, 3, 81 ∗-algebra, 4 Banach-Steinhaus theorem, 175 Borel functional calculus, 41, 95, 135, 142 C∗ -algebra, 4 canonical form of a compact operator, 66 Cayley transform, 148 closable operator, 103 closed graph theorem, 183 operator, 102 closure of an operator, 103, 106 compact operator, 59 containment of operators, 103, 145 continuous functional calculus, 15, 56, 91, 93, 95, 129 core for an operator, 113 cyclic vector, 39 deficiency index, 147 subspace, 147 domain of an operator, 101 Dynkin’s theorem, 42, 137, 177 eigenspaces of a compact operator, 61 eigenvalues of a compact operator, 65 ε-net, 60 essentially self-adjoint operator, 153 essential range, 35 expectation value, 22, 156 extended spectrum, 111 extension Friedrichs, 162 Krein, 162

of an operator, 103 self-adjoint, 147, 151 symmetric, 107 final projection, 26 subspace, 26 finite dimensional operator, 59 formula Lie-Trotter, 51 polarization, ix resolvent, 5, 48, 89 Trotter, 170 Fredholm alternative, 62, 64 Friedrichs extension, 162 Fuglede’s theorem, 53 functional calculus Borel for normal operators, 95 for self-adjoint operators, 41, 135, 142 continuous for elements of a C∗ -algebra, 56 for families of operators, 91, 93 for normal operators, 95 for self-adjoint operators, 15, 129 holomorphic, 47 for elements of a C∗ -algebra, 56 for families of operators, 87 function vanishing at infinity, 191 Gelfand-Naimark theorem, 96 generator of a one-parameter group, 170 graph of adjoint operator, 105 norm, 104 of an operator, 102, 183 group one-parameter, 165 generator of, 170 U(n), 153 Hilbert-Schmidt norm, 78

© Springer Nature Switzerland AG 2018 P. Sołtan, A Primer on Hilbert Space Operators, Compact Textbooks in Mathematics, https://doi.org/10.1007/978-3-319-92061-0

198

Index

operator, 72 on L2 , 84 theorem, 65 holomorphic functional calculus, 47, 56, 87 ∗-homomorphism, 40 horizontal vector, 103 ideal, 59, 186 left, 73, 186 right, 73, 186 two-sided, 186 imaginary part of an operator, 93 infinitesimal generator, 170 initial projection, 26 subspace, 26 integral kernel, 85 operator, 85 invertible element of a C∗ -algebra, 11 operator, 4 unbounded operator, 110 involution, 3, 154 isometry, 27 partial, 26 ∗-isomorphism, 15 joint spectrum, 93 Krein extension, 162 λ-system, 177 left ideal, 73, 186 left support of an operator, 25 Lie-Trotter formula, 51 measure push-forward of, 143 semifinite, 34 spectral, 43, 137 of a normal operator, 96 of a self-adjoint operator, 43, 46 total variation of, 43 variation of, 137 modulus of an operator, 28, 126 multiplication operator, 34, 104 norm, 36 spectrum of, 35, 134

norm graph, 104 Hilbert-Schmidt, 78 of a multiplication operator, 36 quotient, 184 trace, 82 normal element of a C∗ -algebra, 55 operator, 13 one-parameter group, 165 generator of, 170 open mapping theorem, 181 operator anti-linear, 153 anti-unitary, 154 bounded absolute value of, 28 left support of, 25 modulus of, 28 phase of, 28 polar decomposition of, 28 resolvent of, 5 right support of, 25 self-adjoint, 13 spectral radius of, 10 spectrum of, 4 closable, 103 closed, 102 closure of, 103, 106 compact, 59 canonical form of, 66 diagonalization, 65 eigenspaces of, 61 eigenvalues of, 65 spectrum of, 64 densely defined, 102 domain of, 101 essentially self-adjoint, 153 extension of, 103 finite dimensional, 59 graph of, 102 Hilbert-Schmidt, 72 on L2 , 84 integral, 85 invertible, 4 of multiplication, 34, 104 norm, 36 spectrum of, 35

199 Index

normal, 13 spectral measure of, 96 positive, 19, 21, 123 square root of, 20 self-adjoint, 13, 105 negative part, 21 positive part, 21 real and imaginary parts, 93 spectral measure of, 43, 46, 142 support of, 25 similar, 54 symmetric, 105, 147 Cayley transform of, 148 deficiency indices of, 147 deficiency subspaces of, 147 self-adjoint extension of, 147, 151 trace class, 72 unbounded, 101 absolute value of, 126 adjoint of, 105 core for, 113 invertible, 110 modulus of, 126 phase of, 126 polar decomposition of, 126 positive, 123 resolvent set of, 110 self-adjoint, 105 spectrum of, 110 z-transform of, 117 unitary, 13 unitarily equivalent, 39 partial isometry, 26 partial order on B(H), 22 on positive self-adjoint operators, 158 phase of a bounded operator, 28 of an unbounded operator, 126 π-system, 177 polar decomposition of a bounded operator, 28 of an unbounded operator, 126 polarization formula, ix positive element of a C∗ -algebra, 57 operator, 19, 21, 123 projection, 24 initial, final, 26 ψ, ix

ψ , ix push-forward of measure, 143 Putnam’s theorem, 54 quotient Banach space, 184, 185 C∗ -algebra, 189 norm, 184 real part of an operator, 93 resolvent identity, formula, 5, 48, 89 set of a bounded operator, 4 of an unbounded operator, 110 Riesz-Schauder theorem, 64 right ideal, 73, 186 right support of an operator, 25 scalar product, ix self-adjoint element of a C∗ -algebra, 55 operator, 13, 105 self-adjoint extension, 147, 151 semifinite measure space, 34 singular value, 67 spectral mapping theorem, 17, 91, 95 measure, 43, 137 of a normal operator, 96 of a self-adjoint operator, 43, 46, 142 radius, 10 spectrum of a bounded operator, 4 of a compact operator, 64 of an element of a C∗ -algebra, 11 extended, 111 joint, 93 of multiplication operator, 35, 134 square root of a positive operator, 20 Stone’s theorem, 167 strong topology, 29 subspace initial, final, 26 support of a self-adjoint operator, 25 symmetric operator, 105, 147 deficiency indices of, 147 subspaces of, 147

200

Index

tensor product algebraic, 178 of Hilbert spaces, 180 theorem Banach-Steinhaus, 175 closed graph, 183 Dynkin’s, 42, 137, 177 Fuglede’s, 53 Gelfand-Naimark, 96 Hilbert-Schmidt, 65 open mapping, 181 Putnam’s, 54 Riesz-Schauder, 64 spectral mapping, 17, 91, 95 Stone’s, 167 total variation of a measure, 43 trace, 69 class operator, 72 norm, 82 of a positive operator, 71 of a trace class operator, 76 Trotter formula, 170 two-sided ideal, 186

unit approximate, 188 of a C∗ -algebra, 3 unital homomorphism, 40 unitary element of a C∗ -algebra, 55 equivalence, 39 operator, 13 value expectation, 22, 156 singular, 67 variation of a measure, 137 total, 43 vector cyclic, 39 horizontal, 103 vertical, 102 vertical vector, 102 z-transform, 117

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