Fredholm and local spectral theory, with applications to multipliers

Pietro Aiena1402018304, 9781402018305, 9781402025259

“The main aim of this book is to show how the interplay between Fredholm theory and local spectral theory is significant and beautiful. The author begins with the Kato decomposition for bounded operators on Banach spaces and develops, quite systematically, the properties of some important subspaces which play a significant role in Fredholm theory and local spectral theory. A relevant part of the book concerns the single-valued extension property, through which many of the classical results of Fredholm theory are revisited. Another part of the book is devoted to the Fredholm theory of multipliers of commutative semi-simple Banach algebras, and it is shown that in the framework of multiplier theory, the Gelfand theory, Fredholm theory, local spectral theory, and harmonic analysis are closely intertwined. The concluding part of the book regards some recent progress in the study of perturbation theory for some classes of operators which occur in Fredholm theory. This book is intended to be used in graduate courses in operator theory, so the prerequisites are a background in functional and complex analysis. It will be a useful companion for professional mathematicians working in the subject, or interested in applying it to other areas of analysis.