A signi?cant sector of the development of spectral theory outside the classical area of Hilbert space may be found amongst at multipliers de?ned on a complex commutative Banach algebra A. Although the general theory of multipliers for abstract Banach algebras has been widely investigated by several authors, it is surprising how rarely various aspects of the spectral theory, for instance Fredholm theory and Riesz theory, of these important classes of operators have been studied. This scarce consideration is even more surprising when one observes that the various aspects of spectral t- ory mentioned above are quite similar to those of a normal operator de?ned on a complex Hilbert space. In the last ten years the knowledge of the spectral properties of multip- ers of Banach algebras has increased considerably, thanks to the researches undertaken by many people working in local spectral theory and Fredholm theory. This research activity recently culminated with the publication of the book of Laursen and Neumann [214], which collects almost every thing that is known about the spectral theory of multipliers.

From the reviews of the first edition:

"The primary goal of this monograph is a presentation of the Fredholm and Riesz theory of Banach space operators and applications in the stetting of multipliers of a commutative Banach algebra. This book complements standard references for Fredholm theory on the one hand, and Laursen and Neumann's book on the other hand. It should prove to be a valuable resource for graduate students and researchers in Banach space operator theory." (Thomas Len Miller, Mathematical Reviews, 2005e)

"The main concern of the monograph under review is Fredholm theory and its connections with the local spectral theory for bounded linear operators in Banach spaces. The monograph is intended for the use of researchers and graduate students in functional analysis, having a certain background in operator theory. The style is alert and pleasant and there is a fair and state-of-the-art account of the actual Fredholm theory in connection with local spectral theory." (Florian-Horia Vasilescu, Zentralblatt MATH,Vol. 1077 (3), 2006)

Inhalt
The Kato decomposition property.- The single-valued extension property.- The SVEP and Fredholm theory.- Multipliers of commutative Banach algebras.- Abstract Fredholm theory.- Decomposability.- Perturbation classes of operators.