Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras, Second Edition

摘要

Spectral theory is one of the most important domains of modern functional analysis with a wealth of applications in mathematics and physics as well, ranging from matrix theory, complex analysis and differential and integral equations to control theory and quantum physics. This monograph is an axiomatized survey of the major classes of spectra under the unifying concept of regularity. In a Banach algebra A, a regularity R is a subset of A defined in an axiomatic way that enjoys some of the main features of the set of invertible elements of A. These axioms are chosen in such a way so that on one hand, they allow the study of a large class of concrete examples and, on the other hand, are sufficiently strong so that they allow the construction of a consistent theory. A notion of joint regularity is also introduced in order to handle the Taylor spectra. The book has five chapters treating the following topics: Banach algebras, linear operators and their spectra, essential spectrum and perturbations, Taylor spectrum and, finally, orbits and capacity. Each chapter contains a very clever and detailed presentation of the topic with numerous illuminating examples, applications and historical comments. This book is a beautiful and significant contribution to the field of spectral theory.