Spectral Theory of Self-Adjoint Operators in Hilbert Space Mathematics and Its Applications (Soviet Series)
Managing Editor:
M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. A. KIRILLOV, MGV, Moscow, U.S.S.R. Yu. I. MANIN, Steklov Inst. of Math., Moscow, U.S.S.R. N. N. MOISEEV, Camp. Centre, Acad. of Sci., Moscow, U.S.S.R. S. P. NOVIKOV, Landau Inst. Theor. Phys., Moscow, U.S.S.R. M. C. POL YVANOV, Steklov lnst. of Math., Moscow, U.S.S.R. Yu. A. ROZANOV, Steklov lnst. of Math., Moscow, U.S.S.R. M. S. Birman Department of Physics, Leningrad University, U.S.S.R. and M. Z. Solomjak Department of Mathematics, Leningrad University, U.S.S.R.
Spectral Theory of Self-Adjoint Operators in Hilbert Space
D. Reidel Publishing Company
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
Dordrecht / Boston / Lancaster / Tokyo Library of Congress Cataloging-in-Publication Data
Birman, M. Sh. Spectral theory of self-adjoint operators in Hilbert space.
(Mathematics and its applications (Soviet series» Translation of: Spektral'naiii teoriiii samosopriiiz- hennykh operatorov v gil'bertovom prostranstve. Bibliography: p. Includes index. 1. Selfadjoint operators. 2. Spectral theory (Mathematics) 3. Hilbert space. I. Solomiak, M. Z. II. Title. III. Series: Mathematics and its applications (D. Reidel Publishing Company). Soviet series. QA329.2.B5713 1986 515.7'246 86-15645 ISBN-13: 978-94-0 I 0-9009-4 e-ISBN-13: 978-94-009-4586-9 DOT: 10.1007/978-94-009-4586-9
Published by D. Reidel Publishing Company P.O. Box 17,3300 AA Dordrecht, Holland
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Original title: CIIEKTPAJIbHA51 TEOPH51 CAMOCOIIP51iKEHHbIX OIIEPATOPOB B rHJIbBEPTOBOM IIPOCTPAHCTBE Translated from the Russian by S. Khrushchev and V. Peller
All Rights Reserved © 1987 by D. Reidel Publishing Company, Dordrecht, Holland Softcover reprint of the hardcover 1st edition 1987 Original © 1980 by Leningrad University Press. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Contents
Series Editor's Preface lX
~a ~
Preface to the English Edition XIll Basic Notation xv
Chapter 1. Preliminaries 1 1. Metric Spaces. Normed Spaces 1 2. Algebras and a-Algebras of Sets 2 3. Countably Additive Functions and Measures 3 4. Measurable Functions 7 5. Integration 9 6. Function Spaces 14
Chapter 2. Hilbert Space Geometry. Continuous Linear Operators 18 1. Hilbert Space. The Space L2 18 2. Orthonormal Systems 22 3. Projection Theorem. Orthogonal Expansions and Orthogonal Sums 26 4. Linear Functionals and Sesqui-linear Forms. Weak Convergence 32 5. The Algebra of Continuous Operators on H 36 6. Compact Operators 39 7. Bounded Self-adjoint Operators 41 8. Orthogonal Projections 44 9. Examples of Hilbert Spaces and Orthonormal Systems 47 10. Examples of Continuous Functionals and Operators 52
Chapter 3. Unbounded Linear Operators 60 1. General Notions. Graph of an Operator 60 2. Closed Operators. Closable Operators 64 3. Adjoint Operator 68 4. Domination of Operators 72 5. Invariant Subspaces 75 6. Reducing Subsp~ces 78 7. Defect Number, Spectrum, and Resolvent of a Closed Operator 81 8. Skew Decompositions. Skew Reducibility 86 v VI CONTENTS 9. Spectral Theory of Compact Operators 89 10. Connection between the Spectral Properties of TS and ST 93
Chapter 4. Symmetric and Isometric Operators 97 1. Symmetric and Self-adjoint Operators. Deficiency Indices 97 2. Isometric and Unitary Operators 100 3. Cayley Transform 103 4. Extensions of Symmetric Operators. Von Neumann's Formulae 105 5. The Operator T*T. Normal Operators 108 6. Classification of Spectral Points 110 7. Multiplication by the Independent Variable 112 8. Differentiation Operator 116
Chapter 5. Spectral Measure. Integration 123 1. Basic Notions 123 2. Extension of a Spectral Measure. Product Measures 126 3. Integral with Respect to a Spectral Measure. Bounded Functions 130 4. Integral with Respect to a Spectral Measure. Unbounded Functions 133 5. An Example of Commuting Spectral Measures whose Product is not Countably Additive 138
Chapter 6. Spectral Resolutions 140 1. Statements of Spectral Theorems. Functions of Operators 140 2. Spectral Theorem for Unitary Operators 145 3. Spectral Theorem for Self-adjoint,Operators 148 4. Spectral Resolution of a One-parameter Unitary Group 150 5. Joint Spectral Resolution for a Finite Family of Commuting Self-adjoint Operators 153 6. Spectral Resolutions of Normal Operators 156
Chapter 7. Functional Model and the Unitary Invariants of Self-adjoint Operators 159 1. Direct Integral of Hilbert Spaces 159 2. Multiplication Operators and Decomposable Operators 164 3. Generating Systems and Spectral Types 169 4. Unitary Invariants of Spectral Measure 173 5. Unitary Invariants of Self-adjoint Operators 176 6. Decomposition of a Spectral Measure into the Absolutely Continuous and the Singular Part 179
Chapter 8. Some Applications of Spectral Theory 183 1. Polar Decomposition of a Closed Operator 183 2. Differential Equations of Evolution on Hilbert Space 181 CONTENTS vii
3. Fourier Transform 192 4. Multiplications on L2 (Rm , em) 194 5. Differential Operators with Constant Coefficients 197 6. Examples of Differential Operators 201
Chapter 9. Perturbation Theory 206 1. Essential Spectrum. Compact Perturbations 206 2. Compact Self-adjoint and Normal Operators 209 3. Finite-dimensional Perturbations and Extensions 214 4. Continuous Perturbations 217
Chapter 10. Semibounded Oper. 'ors and Forms 221 1. Closed Positive Definite Forms 221 2. Semibounded Forms 224 3. Friedrichs Method of Extension of a Semibounded Operator to a Self-adjoint Operator 228 4. Fractional Powers of Operators. The Heinz Inequality 231 5. Examples of Quadratic Forms. The Sturm-Liouville Operator on [-1, 1] 233 6. Examples of Quadratic Forms. One-dimensional Schr6dinger Operator 238
Chapter 11. Classes of Compact Operators 242 1. Canonical Representation and Singular Numbers of Compact Operators 242 2. Nuclear Operators. Trace of an Operator 245 3. Hilbert-Schmidt Operators 250 4. Sp Classes 253 5. Additional Information on Singular Numbers of Compact Operators 257 6. Lp Classes 262 7. Lidskii's Theorem 268 8. Examples of Compact Operators 271
Chapter 12. Commutation Relations of Quantum Mechanics 279 1. Statement of the Problem. Auxiliary Material 279 2. Properties of (B)-systems and (C)-systems 283 3. Representations of the Bose Relations. The Case m = 1 287 4. Representations of the Bose Relations. General Case 291 5. Representations of the Canonical Relations 294
References 297 Subject Index 299 Series Editor's Preface
Approach your problems from the right end It isn't that they can't see the solution. It is and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. G.K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders.
Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be com• pletely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non• trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order" , which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. This pro• gramme, Mathematics and Its Applications, is devoted to new eme'rging (sub )dis• ciplines and to such (new) interrelations as exempla gratia: - a central concept which plays an important role in several different mathematical and/or scientific specialized areas; - new applications of the results and ideas from one area of scientific endeavour into another; influences which the results, problems and concepts of one field of enquiry have and have had on the development of another. The Mathematics and Its Applications programme tries to make available a careful selection of books which fit the philosophy outlined above. With such books, which are stimulating rather than definitive, intriguing rather than encyclopaedic, we hope to contribute something towards better communication among the practition• ers in diversified fields,
ix x SERIES EDITOR'S PREFACE
Because of the wealth of scholarly research being undertaken in the Soviet Union, Eastern Europe, and Japan, it was decided to devote special attention to the work emanating from these particular regions, Thus it was decided to start three regional series under the umbrella of the main MIA programme. This volume in the MIA (USSR) series has a rather standard sounding title. It is, however, unusual in several respects. For one thing the emphasis definitely is on unbounded operators. There is also a unique chapter on the quantum mechanical commutation relations. Compared to the original Russian edition the major change consists of supplements about compact non-selfadjoint operators. Soviet mathematics has of course a well-earned and long standing reputation of excellence in functional analysis and operator theory. This book, definitely aimed at fields where the special theory of finite families of differential operators is important, will help to reinforce that reputation.
The unreasonable effectiveness of mathemat• As long as algebra and geometry proceeded ics in science ... along separate paths, their advance was slow and their applications limited. Eugene Wigner But when these sciences joined company they drew from each other fresh vitality and Well, if you know of a better 'ole, go to it. thenceforward marched on at a rapid pace towards perfection. Brm;e Bairnsfather Joseph Louis Lagrange. What is now proved was once only imagined.
William Blake
Bussum, May 1986 Michiel Hazewinkel Preface
For several years the authors have given lectures on the spe,ctral theory of oper• ators on Hilbert space and on related topics of analysis and mathematical physics, in particular on the spectral theory of differential operators. These lectures were addressed to students of the Mathematics and Physics Departments of Leningrad University. The plan for this book has developed as a result of this activity. The idea was to set forth in detail, yet in a relatively compact textbook, enough background of the abstract theory of self-adjoint operators on Hilbert space to provide more advanced courses with this material. The book is intended for students (undergraduate, as well as 'graduate) whose interests lie in analysis, mathematical physics, and theoretical physics. This deter• mined the selection of topics and the character of the exposition. A few words about the idiosyncrasies of the book are now in order. Taking into account the needs of applications (of Quantum Theory in particular), we have enlarged the body of questions traditionally treated as 'the core' of Hilbert space theory, i.e. that part of the theory which is usually considered in the first approach to the subject. Attention is focused on unbounded operators. Spectral theory is developed for a finite family of commuting self-adjoint operators. A description of the unitary invariants of such families is given. The exposition is based on the concept of a spectral measure space. In this connection the construction of direct integrals of Hilbert spaces is considered. Three short chapters dealing with the qualitative perturbation theory of the spectrum (Ch. 9), semi-bounded operators and forms (Ch. 10), von Neumann-Schatten classes Sp (Ch. 11) are included. Chapter 8 contains a section with examples of the spectral analysis for partial differential operators. Of course, we did not aim to replace the known treatise on the spectral theory of differential equations. Without claiming completeness, we mention the books by Yu. M. Berezanskii [2], I. M. Glazman [4], K. Maurin [12]. Chapter 12 deserves particular mention. It deals with the commutation relations of Quantum Mechanics. Apparently, this material has not appeared previously in mathematical (textbook) literature. Nevertheless, it is not only important for i
The present translation of our 1980 Russian text is now offered to the reader with a considerable number of supplementary sections. The greater part of the sup• plements concerns the general theory of compact (but not necessarily self-adjoint) operators. The sections dealing with applications and examples are considerably enlarged. Of course, we have used the opportunity to correct some inaccuracies found in the Russian text. The original list of references contained only textbooks and monographs avail• able in Russian. We now give references to the original publications instead of their Russian translations. Moreover, we include a number of original papers in the list of references which have been used in the preparation of the added material. We thank Reidel Publishing Company for undertaking this edition. We acknowl• edge with gratitude the thorough work of our colleagues S. V. Khrushchev and V. V. Peller in translating our book.
xiii Basic Notation
equal to by definition o empty set R = (-00, 00) set of real numbers R+=[O,+oo) Z set of integers z+ = z n R+ C set of complex numbers Re z, 1m z, i real part, imaginary part of z E C, and complex conju• gate of z T unit circle + direct sum of subsets of a linear space dim dimension of a linear space H Hilbert space vM closure of the linear span of M va Fa:= v(UaFa) va{Ua} closure of the linear span of Ua EB,e orthogonal sum and difference M.l. orthogonal complement to M def L = dim U defect of a subspace L D(T) domain of a linear operator T R(T) range of T N(T) kernel of T B = B(H) space of bounded linear operators on H S"" = S",,(H) space of compact linear operators on H K = K(H) set of finite rank operators on H 1= IH identity operator on H T closure of an operator T T* adjoint operator G(T) graph of T u symbol of commutation of linear operators dT(A) = defR(T - U) defect number of T at A ~(T) resolvent set of T ~(T) quasi-regular set of T a(T) spectrum of a closed operator T a(T) core of the spectrum of T xv xvi BASIC NOTATION ap(n point spectrum of T ac(n continuous spectrum of T ar(n residual spectrum of T fA(T) = (T - ).1)-1 resolvent of T aeCA) essential spectrum of a self-adjoint operator A a