Spectral Theory of Linear Operators : And Spectral Systems in Banach Algebras

by Muller, Vladimir

• ISBN13: 9783764369125
• ISBN10: 3764369124
• eBook ISBN(s): 9783034877886
• Format: Hardcover
• Copyright: 2003-05-01
• Publisher: Springer Verlag

Summary

This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach algebras. It presents a survey of results concerning various types of spectra, both of single and n-tuples of elements. Typical examples are the one-sided spectra, the approximate point, essential, local and Taylor spectrum, and their variants. The theory is presented in a unified, axiomatic and elementary way. Many results appear here for the first time in a monograph. The material is self-contained. Only a basic knowledge of functional analysis, topology, and complex analysis is assumed. The monograph should appeal both to students who would like to learn about spectral theory and to experts in the field. It can also serve as a reference book. The present second edition contains a number of new results, in particular, concerning orbits and their relations to the invariant subspace problem. This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach algebras. It presents a survey of results concerning various types of spectra, both of single and n-tuples of elements. Typical examples are the one-sided spectra, the approximate point, essential, local and Taylor spectrum, and their variants. The theory is presented in a unified, axiomatic and elementary way. Many results appear here for the first time in a monograph. The material is self-contained. Only a basic knowledge of functional analysis, topology, and complex analysis is assumed. The present second edition contains a number of new results, in particular, concerning orbits and their relations to the invariant subspace problem. Due to its very clear style and the careful organization of the material, this truly brilliant book may serve as an introduction into the field, yet it also provides an excellent source of information on specific topics in spectral theory for the working mathematician. Review of the first edition by M. Grosser, Vienna Monatshefte f??r Mathematik Vol. 146, No. 1/2005

Table of Contents

Preface
Banach Algebras
Basic concepts	p. 1
Commutative Banach algebras	p. 16
Approximate point spectrum in commutative Banach algebras	p. 25
Permanently singular elements and removability of spectrum	p. 32
Non-removable ideals	p. 41
Axiomatic spectral theory	p. 50
Spectral systems	p. 57
Basic spectral systems in Banach algebras	p. 65
Operators
Spectrum of operators	p. 81
Operators with closed range	p. 92
Factorization of vector-valued functions	p. 101
Kato operators	p. 112
Central inverses and Saphar operators	p. 124
Local spectrum	p. 130
Essential Spectrum
Compact operators	p. 143
Fredholm and semi-Fredholm operators	p. 149
Construction of Sadovskii/Buoni, Harte, Wickstead	p. 159
Perturbation properties of Fredholm and semi-Fredholm operators	p. 163
Essential spectra	p. 166
Ascent, descent and Browder operators	p. 172
Essentially Kato operators	p. 181
Classes of operators defined by means of kernels and ranges	p. 188
Measures of non-compactness and related operator quantities	p. 202
Taylor Spectrum
Basic properties	p. 219
Split spectrum	p. 227
Some non-linear results	p. 234
Taylor functional calculus	p. 241
Local spectrum for n-tuples of operators	p. 256
k-regular functions	p. 261
Stability of index of complexes	p. 270
Essential Taylor spectrum	p. 275
Orbits and Capacity
Joint spectral radius	p. 287
Capacity	p. 297
Local spectral radius and orbits	p. 302
Weak orbits	p. 315
Kaplansky's type theorems	p. 322
Polynomial orbits and local capacity	p. 324
Banach spaces	p. 339
Analytic vector-valued functions	p. 344
C[superscript [infinity]]-functions	p. 347
Semicontinuous set-valued functions	p. 349
Bibliography	p. 351
Index	p. 373
List of Symbols	p. 379
Table of Contents provided by Blackwell. All Rights Reserved.