This book presents the basic tools of modern analysis within the context of what might be called 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 that requiremastery of the material presented here.

Preface

Chapter 1. Spectral Theory and Banach Algebras

 1.1. Origins of Spectral Theory

 1.2. The Spectrum of an Operator

 1.3. Banach Algebras: Examples

 1.4. The Regular Representation

 1.5. The General Linear Group of A

 1.6. Spectrum of an Element of a Banach Algebra

 1.7. Spectral Radius

 1.8. Ideals and Quotients

 1.9. Commutative Banach Algebras

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

 1.11. Spectral Permanence Theorem

 1.12. Brief on the Analytic Functional Calculus

Chapter 2. Operators on Hilbert Space

 2.1. Operators and Their C*-Algebras

 2.2. Commutative C*-Algebras

 2.3. Continuous Functions of Normal Operators

 2.4. The Spectral Theorem and Diagonalization

 2.5. Representations of Banach *-Algebras

 2.6. Borel Functions of Normal Operators

 2.7. Spectral Measures

 2.8. Compact Operators

 2.9. Adjoining a Unit to a C*-Algebra

 2.10. Quotients of C*-Algebras

Chapter 3. Asymptotics: Compact Perturbations and Fredholm Theory

 3.1. The Calkin Algebra

 3.2. Riesz Theory of Compact Operators

 3.3. Fredholm Operators

 3.4. The Fredholm Index

Chapter 4. Methods and Applications

 4.1. Maximal Abelian yon Neumann Algebras

 4.2. Toeplitz Matrices and Toeplitz Operators

 4.3. The Toeplitz C*-Algebra

 4.4. Index Theorem for Continuous Symbols

 4.5. Some H2 Function Theory

 4.6. Spectra of Toeplitz Operators with Continuous Symbol

 4.7. States and the GNS Construction

 4.8. Existence of States: The Gelfand-Naimark Theorem

Bibliography

Index