A Short Course on Spectral Theory

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Springer Science & Business Media, 2002 - Mathematics - 135 pages
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This book presents the basic tools of modern analysis within the context of what might be called the fundamental problem of operator theory: to c- culate spectra of speci?c operators on in?nite-dimensional spaces, especially operators on Hilbert spaces. The tools are diverse, and they provide the basis for more re?ned methods that allow one to approach problems that go well beyond the computation of spectra; the mathematical foundations of quantum physics, noncommutative K-theory, and the classi?cation of sim- ? ple C -algebras being three areas of current research activity that require mastery of the material presented here. The notion of spectrum of an operator is based on the more abstract notion of the spectrum of an element of a complex Banach algebra. - ter working out these fundamentals we turn to more concrete problems of computing spectra of operators of various types. For normal operators, this amounts to a treatment of the spectral theorem. Integral operators require 2 the development of the Riesz theory of compact operators and the ideal L of Hilbert–Schmidt operators. Toeplitz operators require several important tools; in order to calculate the spectra of Toeplitz operators with continuous symbol one needs to know the theory of Fredholm operators and index, the ? structure of the Toeplitz C -algebra and its connection with the topology of curves, and the index theorem for continuous symbols.
  

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Contents

II
1
IV
5
V
7
VI
11
VII
14
VIII
16
IX
18
X
21
XXII
68
XXIII
75
XXIV
78
XXV
83
XXVI
86
XXVII
92
XXVIII
95
XXIX
101

XI
25
XII
27
XIII
31
XIV
33
XV
39
XVI
46
XVII
50
XVIII
52
XIX
57
XX
59
XXI
64
XXX
102
XXXI
106
XXXII
110
XXXIII
114
XXXIV
118
XXXV
120
XXXVI
122
XXXVII
126
XXXVIII
131
XXXIX
133
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