.

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BK

Providence : American Mathematical Society, [2015]

xviii, 749 stran : ilustrace ; 27 cm

ISBN 978-1-4704-1103-9 (vázáno)

Obsahuje bibliografii na stranách 687-726 a rejstříky

001458380

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 // Contents // viii // §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 Szegö’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 L 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