A short course on spectral theory / William Arveson.

Format:

Book

Author/Creator:

Arveson, William.

Series:

Graduate texts in mathematics ; 209.

Graduate texts in mathematics ; 209

Language:

English

Subjects (All):

Spectral theory (Mathematics).

Physical Description:

x, 135 pages ; 24 cm.

Place of Publication:

New York : Springer, [2002]

Summary:

This book presents the basic tools of modern analysis within the context of the fundamental problem of operator theory: to calculate spectra of specific operators on infinite dimensional spaces, especially operators on Hilbert spaces. The tools are diverse, and they provide the basis for more refined 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 classification of simple C*-algebras being three areas of current research activity which require mastery of the material presented here.

Contents:

Chapter 1. Spectral Theory and Banach Algebras 1

1.1. Origins of Spectral Theory 1

1.2. The Spectrum of an Operator 5

1.3. Banach Algebras: Examples 7

1.4. The Regular Representation 11

1.5. The General Linear Group of A 14

1.6. Spectrum of an Element of a Banach Algebra 16

1.7. Spectral Radius 18

1.8. Ideals and Quotients 21

1.9. Commutative Banach Algebras 25

1.10. Examples: C(X) and the Wiener Algebra 27

1.11. Spectral Permanence Theorem 31

1.12. Brief on the Analytic Functional Calculus 33

Chapter 2. Operators on Hilbert Space 39

2.1. Operators and Their C*-Algebras 39

2.2. Commutative C*-Algebras 46

2.3. Continuous Functions of Normal Operators 50

2.4. The Spectral Theorem and Diagonalization 52

2.5. Representations of Banach *-Algebras 57

2.6. Borel Functions of Normal Operators 59

2.7. Spectral Measures 64

2.8. Compact Operators 68

2.9. Adjoining a Unit to a C*-Algebra 75

2.10. Quotients of C*-Algebras 78

Chapter 3. Asymptotics: Compact Perturbations and Fredholm Theory 83

3.1. The Calkin Algebra 83

3.2. Riesz Theory of Compact Operators 86

3.3. Fredholm Operators 92

3.4. The Fredholm Index 95

Chapter 4. Methods and Applications 101

4.1. Maximal Abelian von Neumann Algebras 102

4.2. Toeplitz Matrices and Toeplitz Operators 106

4.3. The Toeplitz C*-Algebra 110

4.4. Index Theorem for Continuous Symbols 114

4.5. Some H[superscript 2] Function Theory 118

4.6. Spectra of Toeplitz Operators with Continuous Symbol 120

4.7. States and the GNS Construction 122

4.8. Existence of States: The Gelfand-Naimark Theorem 126.

Notes:

Includes bibliographical references (page 131) and index.

ISBN:

0387953000

OCLC:

46908837