This page intentionally left blank CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 106
EDITORIAL BOARD B. BOLLOBAS, W. FULTON, A. KATOK, F. KIRWAN, P. SARNAK, B. SIMON, B. TOTARO
LINEAR OPERATORS AND THEIR SPECTRA
This wide ranging but self-contained account of the spectral theory of non-self-adjoint linear operators is ideal for postgraduate students and researchers, and contains many illustrative examples and exercises. Fredholm theory, Hilbert-Schmidt and trace class operators are discussed, as are one- parameter semigroups and perturbations of their generators. Two chapters are devoted to using these tools to analyze Markov semigroups. The text also provides a thorough account of the new theory of pseudospectra, and presents the recent analysis by the author and Barry Simon of the form of the pseu- dospectra at the boundary of the numerical range. This was a key ingredient in the determination of properties of the zeros of certain orthogonal polynomials on the unit circle. Finally, two methods, both very recent, for obtaining bounds on the eigenvalues of non- self-adjoint Schrödinger operators are described. The text concludes with a description of the surprising spectral properties of the non-self-adjoint harmonic oscillator. CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit: http://www.cambridge.org/series/sSeries.asp?code=CSAM Already published 58 J. McCleary A user’s guide to spectral sequences II 59 P. Taylor Practical foundations of mathematics 60 M. P. Brodmann & R. Y. Sharp Local cohomology 61 J. D. Dixon et al. Analytic pro-P groups 62 R. Stanley Enumerative combinatorics II 63 R. M. Dudley Uniform central limit theorems 64 J. Jost & X. Li-Jost Calculus of variations 65 A. J. Berrick & M. E. Keating An introduction to rings and modules 66 S. Morosawa Holomorphic dynamics 67 A. J. Berrick & M. E. Keating Categories and modules with K-theory in view 68 K. Sato Levy processes and infinitely divisible distributions 69 H. Hida Modular forms and Galois cohomology 70 R. Iorio & V. Iorio Fourier analysis and partial differential equations 71 R. Blei Analysis in integer and fractional dimensions 72 F. Borceaux & G. Janelidze Galois theories 73 B. Bollobás Random graphs 74 R. M. Dudley Real analysis and probability 75 T. Sheil-Small Complex polynomials 76 C. Voisin Hodge theory and complex algebraic geometry, I 77 C. Voisin Hodge theory and complex algebraic geometry, II 78 V. Paulsen Completely bounded maps and operator algebras 79 F. Gesztesy & H. Holden Soliton equations and their algebro-geometric solutions, I 81 S. Mukai An Introduction to invariants and moduli 82 G. Tourlakis Lectures in logic and set theory, I 83 G. Tourlakis Lectures in logic and set theory, II 84 R. A. Bailey Association schemes 85 J. Carlson, S. Müller-Stach & C. Peters Period mappings and period domains 86 J. J. Duistermaat & J. A. C. Kolk Multidimensional real analysis I 87 J. J. Duistermaat & J. A. C. Kolk Multidimensional real analysis II 89 M. Golumbic & A. Trenk Tolerance graphs 90 L. Harper Global methods for combinatorial isoperimetric problems 91 I. Moerdijk & J. Mrcun Introduction to foliations and Lie groupoids 92 J. Kollar, K. E. Smith & A. Corti Rational and nearly rational varieties 93 D. Applebaum Levy processes and stochastic calculus 94 B. Conrad Modular forms and the Ramanujan conjecture 95 M. Schecter An introduction to nonlinear analysis 96 R. Carter Lie algebras of finite and affine type 97 H. L. Montgomery, R. C. Vaughan & M. Schechter Multiplicative number theory I 98 I. Chavel Riemannian geometry 99 D. Goldfeld Automorphic forms and L-functions for the group GL(n,R) 100 M. Marcus & J. Rosen Markov processes, Gaussian processes, and local times 101 P. Gille & T. Szamuely Central simple algebras and Galois cohomology 102 J. Bertoin Random fragmentation and coagulation processes 104 A. Ambrosetti & A. Malchiodi Nonlinear analysis and semilinear elliptic problems 105 T. Tao & V. H. Vu Additive combinatorics LINEAR OPERATORS AND THEIR SPECTRA
E. BRIAN DAVIES CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521866293
© E. Brian Davies 2007
This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2007
ISBN-13 978-0-511-28503-5 eBook (EBL) ISBN-10 0-511-28503-5 eBook (EBL)
ISBN-13 978-0-521-86629-3 hardback ISBN-10 0-521-86629-4 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents
Preface page ix
1 Elementary operator theory 1 1.1 Banach spaces1 1.2 Bounded linear operators 12 1.3 Topologies on vector spaces 19 1.4 Differentiation of vector-valued functions 23 1.5 The holomorphic functional calculus 27
2 Function spaces 35 2.1 Lp spaces 35 2.2 Operators acting on Lp spaces 45 2.3 Approximation and regularization 54 2.4 Absolutely convergent Fourier series 60
3 Fourier transforms and bases 67 3.1 The Fourier transform 67 3.2 Sobolev spaces 77 3.3 Bases of Banach spaces 80 3.4 Unconditional bases 90
4 Intermediate operator theory 99 4.1 The spectral radius 99 4.2 Compact linear operators 102 4.3 Fredholm operators 116 4.4 Finding the essential spectrum 124
v vi Contents
5 Operators on Hilbert space 135 5.1 Bounded operators 135 5.2 Polar decompositions 137 5.3 Orthogonal projections 140 5.4 The spectral theorem 143 5.5 Hilbert-Schmidt operators 151 5.6 Trace class operators 153 5.7 The compactness of f Q g P 160
6 One-parameter semigroups 163 6.1 Basic properties of semigroups 163 6.2 Other continuity conditions 177 6.3 Some standard examples 182
7 Special classes of semigroup 190 7.1 Norm continuity 190 7.2 Trace class semigroups 194 7.3 Semigroups on dual spaces 197 7.4 Differentiable and analytic vectors 201 7.5 Subordinated semigroups 205
8 Resolvents and generators 210 8.1 Elementary properties of resolvents 210 8.2 Resolvents and semigroups 218 8.3 Classification of generators 227 8.4 Bounded holomorphic semigroups 237
9 Quantitative bounds on operators 245 9.1 Pseudospectra 245 9.2 Generalized spectra and pseudospectra 251 9.3 The numerical range 264 9.4 Higher order hulls and ranges 276 9.5 Von Neumann’s theorem 285 9.6 Peripheral point spectrum 287
10 Quantitative bounds on semigroups 296 10.1 Long time growth bounds 296 10.2 Short time growth bounds 300 10.3 Contractions and dilations 307 10.4 The Cayley transform 310 Contents vii
10.5 One-parameter groups 315 10.6 Resolvent bounds in Hilbert space 321
11 Perturbation theory 325 11.1 Perturbations of unbounded operators 325 11.2 Relatively compact perturbations 330 11.3 Constant coefficient differential operators on the half-line 335 11.4 Perturbations: semigroup based methods 339 11.5 Perturbations: resolvent based methods 350
12 Markov chains and graphs 355 12.1 Definition of Markov operators 355 12.2 Irreducibility and spectrum 359 12.3 Continuous time Markov chains 362 12.4 Reversible Markov semigroups 366 12.5 Recurrence and transience 369 12.6 Spectral theory of graphs 374
13 Positive semigroups 380 13.1 Aspects of positivity 380 13.2 Invariant subsets 386 13.3 Irreducibility 390 13.4 Renormalization 393 13.5 Ergodic theory 395 13.6 Positive semigroups on C X 399
14 NSA Schrödinger operators 408 14.1 Introduction 408 14.2 Bounds on the numerical range 409 14.3 Bounds in one space dimension 412 14.4 The essential spectrum of Schrödinger operators 420 14.5 The NSA harmonic oscillator 424 14.6 Semi-classical analysis 427
References 436
Index 446
Preface
This volume is halfway between being a textbook and a monograph. It describes a wide variety of ideas, some classical and others at the cutting edge of current research. Because it is directed at graduate students and young researchers, it often provides the simplest version of a theorem rather than the deepest one. It contains a variety of examples and problems that might be used in lecture courses on the subject. It is frequently said that over the last few decades there has been a decisive shift in mathematics from the linear to the non-linear. Even if this is the case it is easy to justify writing a book on the theory of linear operators. The range of applications of the subject continues to grow rapidly, and young researchers need to have an accessible account of its main lines of development, together with references to further sources for more detailed reading. Probability theory and quantum theory are two absolutely fundamental fields of science. In terms of their technological impact they have been far more important than Einstein’s relativity theory. Both are entirely linear. In the first case this is in the nature of the subject. Many sustained attempts have been made to introduce non-linearities into quantum theory, but none has yet been successful, while the linear theory has gone from triumph to triumph. Nobody can predict what the future will hold, but it seems likely that quantum theory will be used for a long time yet, even if a non-linear successor is found. The fundamental equations of quantum mechanics involve self-adjoint and unitary operators. However, once one comes to applications, the situation changes. Non-self-adjoint operators play an important role in topics as diverse as the optical model of nuclear scattering, the analysis of resonances using complex scaling, the behaviour of unstable lasers and the scattering of atoms by periodic electric fields.1
1 A short list of references to such problems may be found in [Berry, website].
ix x Preface
There are many routes into the theory of non-linear partial differential equations. One approach depends in a fundamental way on perturbing linear equations. Another idea is to use comparison theorems to show that certain non-linear equations retain desired properties of linear cousins. In the case of the Kortweg-de Vries equation, the exact solution of a highly non-linear equation depends on reducing it to a linear inverse problem. In all these cases progress depends upon a deep technical knowledge of what is, and is not, possible in the linear theory. A standard technique for studying the non-linear stochastic Navier-Stokes equation involves reformulating it as a Markov process acting on an infinite-dimensional configuration space X. This process is closely associated with a linear Markov semigroup acting on a space of observables, i.e. bounded functions f X→ C. The decay properties of this semigroup give valuable information about the behaviour of the original non-linear equation. The material in Section 13.6 is related to this issue. There is a vast number of applications of spectral theory to problems in engineering, and I mention just three. The unexpected oscillations of the London Millennium Bridge when it opened in 2000 were due to inadequate eigenvalue analysis. There is a considerable literature analyzing the charac- teristic timbres of musical instruments in terms of the complex eigenvalues of the differential equations that govern their vibrations. Of more practical importance are resonances in turbines, which can destroy them if not taken seriously. As a final example of the importance of spectral theory I select the work of Babenko, Mayer and others on the Gauss-Kuzmin theorem about the distribution of continued fractions, which has many connections with modular curves and other topics; see [Manin and Marcolli 2002]. This profound work involves many different ideas, but a theorem about the dominant eigenvalue of a certain compact operator having an invariant closed cone is at the centre of the theory. This theorem is close to ideas in Chapter 13, and in particular to Theorem 13.1.9. Once one has decided to study linear operators, a fundamental choice needs to be made. Self-adjoint operators on Hilbert spaces have an extremely detailed theory, and are of great importance for many applications. We have carefully avoided trying to compete with the many books on this subject and have concentrated on the non-self-adjoint theory. This is much more diverse – indeed it can hardly be called a theory. Studying non-self-adjoint operators is like being a vet rather than a doctor: one has to acquire a much wider range of knowledge, and to accept that one cannot expect to have as high a rate of success when confronted with particular cases. It comprises a collection of methods, each of which is useful for some class of such operators. Some Preface xi of these are described in the recent monograph of Trefethen and Embree on pseudospectra, Haase’s monograph based on the holomorphic functional calculus, Ouhabaz’s detailed theory of the Lp semigroups associated with NSA second order elliptic operators, and the much older work of Sz.-Nagy and Foias, still being actively developed by Naboko and others. If there is a common thread in all of these it is the idea of using theorems from analytic function theory to understand NSA operators. One of the few methods with some degree of general application is the theory of one-parameter semigroups. Many of the older monographs on this subject (particularly my own) make rather little reference to the wide range of applications of the subject. In this book I have presented a much larger number of examples and problems here in order to demonstrate the value of the general theory. I have also tried to make it more user-friendly by including motivating comments. The present book has a slight philosophical bias towards explicit bounds and away from abstract existence theorems. I have not gone so far as to insist that every result should be presented in the language of constructive analysis, but I have sometimes chosen more constructive proofs, even when they are less general. Such proofs often provide new insights, but at the very least they may be more useful for numerical analysts than proofs which merely assert the existence of a constant or some other entity. There are, however, many entirely non-constructive proofs in the book. The fact that the spectrum of a bounded linear operator is always non-empty depends upon Liouville’s theorem and a contradiction argument. It does not suggest a procedure for finding even one point in the spectrum. It should therefore come as no surprise that the spectrum can be highly unstable under small perturbations of the operator. The pseudospectra are more stable, and because of that arguably more important for non-self-adjoint operators. It is particularly hard to give precise historical credit for many theorems in analysis. The most general version of a theorem often emerges several decades after the first one, with a proof which may be completely different from the original one. I have made no attempt to give references to the original literature for results discovered before 1950, and have attached the conventional names to theorems of that era. The books of Dunford and Schwartz should be consulted for more detailed information; see [Dunford and Schwartz 1966, Dunford and Schwartz 1963]. I only assign credit on a systematic basis for results proved since 1980, which is already a quarter of a century ago. I may not even have succeeded in doing that correctly, and hope that those who feel slighted will forgive my failings, and let me know, so that the situation can be rectified on my website and in future editions. xii Preface
I conclude by thanking the large number of people who have influenced me, particularly in relation to the contents of this book. The most important of these have been Barry Simon and, more recently, Nick Trefethen, to both of whom I owe a great debt. I have also benefited greatly from many discussions with Wolfgang Arendt, Anna Aslanyan, Charles Batty, Albrecht Böttcher, Lyonell Boulton, Ilya Goldsheid, Markus Haase, Evans Harrell, Paul Incani, Boris Khoruzhenko, Michael Levitin, Terry Lyons, Reiner Nagel, Leonid Parnovski, Michael Plum, Yuri Safarov, Eugene Shargorodsky, Stanislav Shkarin, Johannes Sjöstrand, Dan Stroock, John Weir, Hans Zwart, Maciej Zworski and many other good friends and colleagues. Finally I want to record my thanks to my wife Jane, whose practical and moral support over many years has meant so much to me. She has also helped me to remember that there is more to life than proving theorems! 1 Elementary operator theory
1.1 Banach spaces
In this chapter we collect together material which should be covered in an introductory course of functional analysis and operator theory. We do not always include proofs, since there are many excellent textbooks on the subject.1 The theorems provide a list of results which we use throughout the book. We start at the obvious point. A normed space is a vector space (assumed to be over the complex number field C) provided with a norm · satisfying