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Operator Theory A Comprehensive Course in Analysis, Part 4 Barry Simon Operator Theory A Comprehensive Course in Analysis, Part 4 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 informationand updates on this book, visit www.ams.org/bookpages/simon Library of Congress Cataloging-in-Publication Data Simon,Barry,1946– Operatortheory/BarrySimon. pagescm. —(Acomprehensivecourseinanalysis;part4) Includesbibliographicalreferencesandindexes. ISBN978-1-4704-1103-9(alk.paper) 1.Mathematicalanalysis—Textbooks. 2.Operatortheory. I.Title. QA300.S5285 2015 515(cid:2).724—dc23 2015033104 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for useinteachingorresearch. Permissionisgrantedtoquotebriefpassagesfromthispublicationin reviews,providedthecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink(cid:2) service. Formoreinformation,pleasevisit: http://www.ams.org/rightslink. Sendrequestsfortranslationrightsandlicensedreprintstoreprint-permission@ams.org. Excludedfromtheseprovisionsismaterialforwhichtheauthorholdscopyright. Insuchcases, requestsforpermissiontoreuseorreprintmaterialshouldbeaddresseddirectlytotheauthor(s). Copyrightownershipisindicatedonthecopyrightpage,oronthelowerright-handcornerofthe firstpageofeacharticlewithinproceedingsvolumes. (cid:2)c 2015bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://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