Banach Algebra Techniques in Operator Theory

כריכה קדמית
Springer Science & Business Media, 6 בדצמ׳ 2012 - 198 עמודים
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Operator theory is a diverse area of mathematics which derives its impetus and motivation from several sources. It began with the study of integral equations and now includes the study of operators and collections of operators arising in various branches of physics and mechanics. The intention of this book is to discuss certain advanced topics in operator theory and to provide the necessary background for them assuming only the standard senior-first year graduate courses in general topology, measure theory, and algebra. At the end of each chapter there are source notes which suggest additional reading along with giving some comments on who proved what and when. In addition, following each chapter is a large number of problems of varying difficulty. This new edition will appeal to a new generation of students seeking an introduction to operator theory.
 

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תוכן

Banach Algebras
30
Abstract Index in a Banach Algebra
33
The Space of Multiplicative Linear Functions
36
The Gelfand Transform
37
The GelfandMazur Theorem
39
The Gelfand Theorem for Commutative Banach Algebras
41
The Spectral Radius Formula
42
The StoneWeierstrass Theorem
46
Normal Operators with Cyclic Vectors
95
Maximal Abelian WAlgebras
98
Homomorphisms of CAlgebras
100
The Extended Functional Calculus
102
The Fuglede Theorem
103
Notes
104
Compact Operators Fredholm Operators and Index Theory
108
Approximation of Compact Operators
110

The Disk Algebra
47
The Algebra of Functions with Absolutely Convergent Fourier Series
50
The Algebra of Bounded Measurable Functions
52
Notes
53
Geometry of Hilbert Space
58
The CauchySchwarz Inequality
60
Hilbert Spaces
61
The Riesz Representation Theorem
66
The Existence of Orthonormal Bases
69
The Dimension of Hilbert Spaces
70
Notes
71
Operators on Hilbert Space and CAlgebras
74
Normal and Selfadjoint Operators
77
Projections and Subspaces
78
Multiplication Operators and Maximal Abelian Algebras
80
The Bilateral Shift Operator
82
CAlgebras
83
The GelfandNaimark Theorem
84
The Functional Calculus
86
The Unilateral Shift Operator
87
The Polar Decomposition
88
Weak and Strong Operator Topologies
91
WAlgebras
92
Isomorphisms of LSpaces
94
Integral Operators
112
The Calkin Algebra and Fredholm Operators
113
Atkinsons Theorem
114
The Index of Fredholm Operators
115
The Fredholm Alternative
116
Volterra Integral Operators
118
Connectedness of the Unitary Group in a WAlgebra
119
Characterization of Index
123
Quotient CAlgebras
124
Representations of the CAlgebra of Compact Operators
126
Notes
129
The Hardy Spaces
133
Reducing Subspaces of Unitary Operators
135
Beurlings Theorem
137
The Maximal Ideal Space of H
138
The InnerOuter Factorization of Functions in H
141
The Conjugates of H and LH5
144
The Closedness of H + C
145
The GleasonWhitney Theorem
146
Toeplitz Operators
158
References
185
Index
191
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