Operator Theory A Comprehensive Course in Analysis, Part 4 Barry Simon Operator Theory A Comprehensive Course in Analysis, Part 4 http://dx.doi.org/10.1090/simon/004 Operator Theory A Comprehensive Course in Analysis, Part 4 Barry Simon Providence, Rhode Island 2010 Mathematics Subject Classification. Primary 47-01, 34-01, 46-01; Secondary 81Q10, 34L05, 35P05, 42C05, 46H05, 22B05. For additional information and updates on this book, visit www.ams.org/bookpages/simon Library of Congress Cataloging-in-Publication Data Simon, Barry, 1946– Operator theory / Barry Simon. pages cm. — (A comprehensive course in analysis ; part 4) Includes bibliographical references and indexes. ISBN 978-1-4704-1103-9 (alk. paper) 1. Mathematical analysis—Textbooks. 2. Operator theory. I. Title. QA300.S5285 2015 515.724—dc23 2015033104 Copying and reprinting. 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Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2015 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 201918171615 To the memory of Cherie Galvez extraordinary secretary, talented helper, caring person and to the memory of my mentors, Ed Nelson (1932-2014) and Arthur Wightman (1922-2013) who not only taught me Mathematics but taught me how to be a mathematician Contents Preface to the Series xi Preface to Part 4 xvii Chapter 1. Preliminaries 1 §1.1. Notation and Terminology 1 §1.2. Some Complex Analysis 3 §1.3. Some Linear Algebra 6 §1.4. Finite-Dimensional Eigenvalue Perturbation Theory 21 §1.5. Some Results from Real Analysis 28 Chapter 2. Operator Basics 33 §2.1. Topologies and Special Classes of Operators 34 §2.2. The Spectrum 46 §2.3. The Analytic Functional Calculus 58 §2.4. The Square Root Lemma and the Polar Decomposition 71 Chapter 3. Compact Operators, Mainly on a Hilbert Space 89 §3.1. Compact Operator Basics 91 §3.2. The Hilbert–Schmidt Theorem 102 §3.3. The Riesz–Schauder Theorem 111 §3.4. Ringrose Structure Theorems 120 §3.5. Singular Values and the Canonical Decomposition 132 §3.6. The Trace and Trace Class 136 §3.7. Bonus Section: Trace Ideals 145 vii viii Contents §3.8. Hilbert–Schmidt Operators 154 §3.9. Schur Bases and the Schur–Lalesco–Weyl Inequality 161 §3.10. Determinants and Fredholm Theory 164 §3.11. Operators with Continuous Integral Kernels 174 §3.12. Lidskii’s Theorem 184 §3.13. Bonus Section: Regularized Determinants 187 §3.14. Bonus Section: Weyl’s Invariance Theorem 192 §3.15. Bonus Section: Fredholm Operators and Their Index 201 §3.16. Bonus Section: M. Riesz’s Criterion 223 Chapter 4. Orthogonal Polynomials 229 §4.1. Orthogonal Polynomials on the Real Line and Favard’s Theorem 231 §4.2. The Bochner–Brenke Theorem 242 §4.3. L2- and L∞-Variational Principles: Chebyshev Polynomials 256 §4.4. Orthogonal Polynomials on the Unit Circle: Verblunsky’s and Szeg˝o’s Theorems 268 Chapter 5. The Spectral Theorem 287 §5.1. Three Versions of the Spectral Theorem: Resolutions of the Identity, the Functional Calculus, and Spectral Measures 289 §5.2. Cyclic Vectors 301 §5.3. A Proof of the Spectral Theorem 301 §5.4. Bonus Section: Multiplicity Theory 303 §5.5. Bonus Section: The Spectral Theorem for Unitary Operators 316 §5.6. Commuting Self-adjoint and Normal Operators 323 §5.7. Bonus Section: Other Proofs of the Spectral Theorem 328 §5.8. Rank-One Perturbations 333 §5.9. Trace Class and Hilbert–Schmidt Perturbations 345 Chapter 6. Banach Algebras 355 §6.1. Banach Algebra: Basics and Examples 357 §6.2. The Gel’fand Spectrum and Gel’fand Transform 370 §6.3. Symmetric Involutions 392 §6.4. Commutative Gel’fand–Naimark Theorem and the Spectral Theorem for Bounded Normal Operators 400 Contents ix §6.5. Compactifications 407 §6.6. Almost Periodic Functions 413 §6.7. The GNS Construction and the Noncommutative Gel’fand–Naimark Theorem 421 §6.8. Bonus Section: Representations of Locally Compact Groups 430 §6.9. Bonus Section: Fourier Analysis on LCA Groups 448 §6.10. Bonus Section: Introduction to Function Algebras 469 §6.11. Bonus Section: The L1(R) Wiener and Ingham Tauberian Theorems 493 §6.12. The Prime Number Theorem via Tauberian Theorems 510 Chapter 7. Bonus Chapter: Unbounded Self-adjoint Operators 515 §7.1. Basic Definitions and the Fundamental Criterion for Self-adjointness 518 §7.2. The Spectral Theorem for Unbounded Operators 541 §7.3. Stone’s Theorem 549 §7.4. von Neumann’s Theory of Self-adjoint Extensions 554 §7.5. Quadratic Form Methods 572 §7.6. Pointwise Positivity and Semigroup Methods 610 §7.7. Self-adjointness and the Moment Problem 633 §7.8. Compact, Rank-One and Trace Class Perturbations 660 §7.9. The Birman–Schwinger Principle 668 Bibliography 687 Symbol Index 727 Subject Index 729 Author Index 741 Index of Capsule Biographies 749 Preface to the Series Young men should prove theorems, old men should write books. —Freeman Dyson, quoting G. H. Hardy1 Reed–Simon2 starts with “Mathematics has its roots in numerology, geom- etry, and physics.” This puts into context the division of mathematics into algebra, geometry/topology, and analysis. There are, of course, other ar- eas of mathematics, and a division between parts of mathematics can be artificial. But almost universally, we require our graduate students to take courses in these three areas. This five-volume series began and, to some extent, remains a set of texts for a basic graduate analysis course. In part it reflects Caltech’s three-terms- per-year schedule and the actual courses I’ve taught in the past. Much of the contents of Parts 1 and 2 (Part 2 is in two volumes, Part 2A and Part 2B) are common to virtually all such courses: point set topology, measure spaces, Hilbert and Banach spaces, distribution theory, and the Fourier transform, complex analysis including the Riemann mapping and Hadamard product theorems. Parts 3 and 4 are made up of material that you’ll find in some, but not all, courses—on the one hand, Part 3 on maximal functions and Hp spaces; on the other hand, Part 4 on the spectral theorem for bounded self-adjoint operators on a Hilbert space and det and trace, again for Hilbert space operators. Parts 3 and 4 reflect the two halves of the third term of Caltech’s course. 1Interview with D. J. Albers, The College Mathematics Journal, 25, no. 1, January 1994. 2M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New York, 1972. xi xii Preface to the Series While there is, of course, overlap between these books and other texts, there are some places where we differ, at least from many: (a) By having a unified approach to both real and complex analysis, we are able to use notions like contour integrals as Stietljes integrals that cross the barrier. (b) We include some topics that are not standard, although I am sur- prised they are not. For example, while discussing maximal functions, I present Garcia’s proof of the maximal (and so, Birkhoff) ergodic the- orem. (c) These books are written to be keepers—the idea is that, for many stu- dents, this may be the last analysis course they take, so I’ve tried to write in a way that these books will be useful as a reference. For this reason, I’ve included “bonus” chapters and sections—material that I do not expect to be included in the course. This has several advantages. First, in a slightly longer course, the instructor has an option of extra topics to include. Second, there is some flexibility—for an instructor who can’t imagine a complex analysis course without a proof of the prime number theorem, it is possible to replace all or part of the (non- bonus) chapter on elliptic functions with the last four sections of the bonus chapter on analytic number theory. Third, it is certainly possible to take all the material in, say, Part 2, to turn it into a two-term course. Most importantly, the bonus material is there for the reader to peruse long after the formal course is over. (d) I have long collected “best” proofs and over the years learned a num- ber of ones that are not the standard textbook proofs. In this re- gard, modern technology has been a boon. Thanks to Google books and the Caltech library, I’ve been able to discover some proofs that I hadn’t learned before.