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.

Inhalt

Preface 1. The Kato decomposition property 1. Hyper-kernel and hyper-range of an operator 2. Semi-regular operators on Banach spaces 3. Analytical core of an operator 4. The semi-regular spectrum of an operator 5. The generalized Kato decomposition 6. Semi-Fredholm operators 7. Quasi-nilpotent part of an operator 2. The single-valued extension property 1. Local spectrum and SVEP 2. The SVEP at a point 3. A local spectral mapping theorem 4. Algebraic spectral subspaces 5. Weighted shift operators and SVEP 3. The SVEP and Fredholm theory 1. Ascent, descent, and the SVEP 2. The SVEP for operators of Kato type 3. The SVEP on the components of rho kappa (T) 4. The Fredholm, Weyl, and Browder spectra 5. Compressions 6. Some spectral mapping theorems 7. Isolated points of the spectrum 8. Weyl's theorem 9. Riesz operators 10. The spectra of some operators 4. Multipliers of commutative Banach algebras 1. Definitions and elementary properties 2. The Helgason-Wang function 3. The first spectral properties of multipliers 4. Multipliers of group algebras 5. Multipliers of Banach algebras with orthogonal basis 6. Multipliers of commutative H* algebras 5. Abstract Fredholm theory 1. Inessential ideals 2. The socle 3. The socle of semi-prime Banach algebras 4. Riesz algebras 5. Fredholm elements of Banach algebras 6. Compact multipliers 7. Weyl multipliers 8. Multipliers of Tauberian regular commutative algebras 9. Some concrete cases 10. Browder spectrum of a multiplier 6. Decomposability 1. Spectral maximal subspaces 2. Decomposable operators on Banach spaces 3. Super-decomposable operators 4. Decomposable right shift operators 5. Decomposable multipliers 6. Riesz multipliers 7. Decomposable convolution operators 7.Perturbation classes of operators 1. Inessential operators between Banach spaces 2. Omega+ and Omega- operators 3. Strictly singular and strictly cosingular operators 4. Improjective operators 5. Incomparability between Banach spaces Bibliography Index