算子理论是现代数学的许多重要领域的重要组成部分，这些领域包括：泛函分析、微分方程、指标理论、表示论和数学物理等等。本书内容涵盖算子理论的中心课题，并以极好的清晰度和风格进行讲述，使读者可以联想到 Conway 的写作风格。 前面几章介绍并回顾了C -代数、正规算子、紧算子和非紧算子。部分主要论题包含了谱理论、泛函演算和 Fredholm 指标。另外，还论述了算子理论和解析函数间某些深刻的联系。 后面几章讲述了更高级的专题，包括了诸如 C -代数的表示、紧微扰和 von Neumann 代数。同样讲述了诸如 Sz.-Nagy 伸缩定理、Weyl-Fredholm 定理和 von Neumann 代数分类等重要结果，以及对Fredholm 理论的处理。最后一章介绍了自返子空间，它连同超自反空间是非对称代数的现代研究中诸多成功的事件之一。
Preface
Chapter 1．Introduction to C*-Algebras
1．Definition and examples
2．Abelian C*-algebras and the Functional Calculus
3．The positive elements in a C*-algebra
4．Approximate identities
5．Ideals in a C*-algebra
6．Representations of a C*-algebra
7．Positive linear functionals and the GNS construction
Chapter 2．Normal Operators
8．Some topologies on B（H）
9．Spectral measures
10．The Spectral Theorem
11．Star-cyclic normal operators
12．The commutant
13．Von Neumann algebras
14．Abelian von Neumann algebras
15．The functional calculus for normal operators
Chapter 3．Compact Operators
16．C*-algebras of compact operators
17．Ideals of operators
18．Trace class and Hilbert-Schmidt operators
19．The dual spaces of the compact operators and the trace class
20．The weak-star topology
21．Inflation and the topologies
Chapter 4．Some Non-Normal Operators
22．Algebras and lattices
23．Isometries
24．Unilateral and bilateral shifts
25．Some results on Hardy spaces
26．The functional calculus for the unilateral shift
27．Weighted shifts
28．The Volterra operator
29．Bergman operators
30．Subnormal operators
31．Essentially normal operators
Chapter 5．More on C*-Algebras
32．Irreducible representations
33．Positive maps
34．Completely positive maps
35．An application：Spectral sets and the Sz．-Nagy DilationTheorem
36．Quasicentral approximate identitites
Chapter 6．Compact Perturbations
37．Behavior of the spectrum under a compact perturbation
38．Bp perturbations of hermitian operators
39．The Weyl-von Neumann-Berg Theorem
40．Voiculescu's Theorem
41．Approximately equivalent representations
42．Some applications
Chapter 7．Introduction to Von Neumann Algebras
43．Elementary properties and examples
44．The Kaplansky Density Theorem
45．The Pedersen Up-Down Theorem
46．Normal homomorphisms and ideals
47．Equivalence of projections
48．Classification of projections
49．Properties of projections
50．The structure of Type I algebras
51．The classification of Type I algebras
52．Operator-valued measurable functions
53．Some structure theory for continuous algebras
54．Weak-star continuous linear functionals revisited
55．The center-valued trace
Chapter 8．Reflexivity
56．Fundamentals and examples
57．Reflexive operators on finite dimensional spaces
58．Hyperreflexive subspaces
59．Reflexivity and duality
60．Hypereflexive von Neumann algebras
61．Some examples of operators
Bibliography
Index
List of Symbols
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