算子理论: 分析综合教程(第4部分)(影印版)

Poincare奖得主 BarrySimon的《分析综合教 程》是一套五卷本的经典教 程，可以作为研究生阶段的 分析学教科书。这套分析教 程提供了很多额外的信息， 包含数百道习题和大量注释 ，这些注释扩展了正文内容 并提供了相关知识的重要历 史背景。阐述的深度和广度 使这套教程成为几乎所有经 典分析领域的宝贵参考资料 。 第4部分侧重于算子理论 ，尤其是Hilbert空间。中心 主题是谱定理、迹类理论和 Fredholm行列式，以及无 界自伴算子的研究。此外还 介绍了正交多项式理论和关 于Banach代数的长章，包 括交换和非交换Gel'fand- Naimark定理以及对一般局 部紧致Abel群的Fourier分析 。 本书可供专业研究人员 （数学家、部分应用数学家 和物理学家）、讲授研究生 阶段分析课程的教师以及在 工作和学习中需要任何分析 学知识的研究生阅读参考。

Preface to the Series Preface to Part 4 Chapter 1. Preliminaries §1.1. Notation and Terminology §1.2. Some Complex Analysis §1.3. Some Linear Algebra §1.4. Finite-Dimensional Eigenvalue Perturbation Theory §1.5. Some Results from Real Analysis Chapter 2. Operator Basics §2.1. Topologies and §pecial Classes of Operators §2.2. The Spectrum §2.3. The Analytic Functional Calculus §2.4. The Square Root Lemma and the Polar Decomposition Chapter 3. Compact Operators, Mainly on a Hilbert §pace §3.1. Compact Operator Basics §3.2. The Hilbert-§chmidt Theorem §3.3. The Riesz-Schauder Theorem §3.4. Ringrose Structure Theorems §3.5. Singular Values and the Canonical Decomposition §3.6. The Trace and Trace Class §3.7. Bonus Section: Trace Ideals §3.8. Hilbert-Schmidt Operators §3.9. Schur Bases and the Schur-Lalesco-Weyl Inequality §3.10. Determinants and Fredholm Theory §3.11. Operators with Continuous Integral Kernels §3.12. Lidski's Theorem §3.13. Bonus Section: Regularized Determinants §3.14. Bonus Section: Weyl's Invariance Theorem §3.15. Bonus Section: Fredholm Operators and Their Index §3.16. Bonus Section: M. Riesz's Criterion Chapter 4. Orthogonal Polynomials §4.1. Orthogonal Polynomials on the Real Line and Favard's Theorem §4.2. The Bochner-Brenke Theorem §4.3. L2-and L∞-Variational Principles: Chebyshev Polynomials §4.4. Orthogonal Polynomials on the Unit Circle: Verblunsky's and Szego's Theorems Chapter 5. The Spectral Theorem §5.1. Three Versions of the Spectral Theorem: Resolutions of the Identity, the Functional Calculus, and Spectral Measures §5.2. Cyclic Vectors §5.3. A Proof of the Spectral Theorem §5.4. Bonus Section: Multiplicity Theory §5.5. Bonus Section: The Spectral Theorem for UnitaryOperators §5.6. Commuting Self-adjoint and Normal Operators §5.7. Bonus Section: Other Proofs of the Spectral Theorem §5.8. Rank-One Perturbations §5.9. Trace Class and Hilbert-Schmidt Perturbations Chapter 6. Banach Algebras §6.1. Banach Algebra: Basics and Examples §6.2. The Gel'fand Spectrum and Gel'fand Transform §6.3. Symmetric Involutions §6.4. Commutative Gel'fand-Naimark Theorem and the Spectral Theorem for Bounded Normal Operators