##
**Fredholm and local spectral theory, with applications to multipliers.**
*(English)*
Zbl 1077.47001

Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1830-4/hbk). xiv, 444 p. (2004).

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 author endeavors to present the important progress made in the last years in the study of the perturbation theory of Banach space operators. In particular, recent solutions to some old open problems are discussed.

The first chapter of the monograph deals with the Kato decomposition property, which holds for a class of linear operators containing the semi-Fredholm operators. The concepts of semi-regular, essentially semi-regular and Kato type operators are introduced, all of them leading to distinguished spectra.

The second chapter is devoted to a localized version of the single-valued extension property at a point, while the third chapter describes the connections between local spectral theory and Fredholm theory.

The fourth chapter is devoted to the theory of multipliers of commutative semi-simple Banach algebras. Desiring to make the work as much self-contained as possible, the elementary theory of multipliers is included. Although some preliminary results are investigated in the non-commutative case, the author focuses his attention on commutative algebras, where the Gelfand theory is available.

The introduction of basic facts concerning the abstract Fredholm theory in arbitrary Banach algebras forms the first part of the fifth chapter. An important role in the discussion is played by the concepts of inessential ideal and the socle in a Banach algebra. We recall that the inessential ideal of the socle corresponds in the case of the Banach algebra of all bounded linear operators to the ideal of all bounded finite rank operators.

Riesz algebras, compact multipliers and Weyl multipliers are also studied in the fifth chapter. The theory of decomposable operators, introduced by C. Foiaş in 1963, is the main subject of the sixth chapter. Using various simplifications of the original definition of Foiaş, and after the presentation of their general properties, the author deals with decomposable operators belonging to several classes: right shift operators, multipliers, convolution operators, etc.

The seventh and last chapter of this monograph is devoted to the study of some perturbation classes of operators as, for instance, the classes of Fredholm operators, upper and lower semi-Fredholm operators, and various ideals of operators. Topics such as inessential operators between Banach spaces, strictly singular and strictly cosingular operators, and improjective operators are also approached in this chapter.

This 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. Unfortunately, a slight carelessness can be sometimes detected. For instance, regarding the introduction of decomposability (see the author’s comment on page 309 of his book), due to C. Foiaş, the author prefers to mention the monograph quoted as [83] [see I. Colojoara and C. Foias, “Theory of generalized spectral operators” (Mathematics and its Applications 9) (Gordon and Breach Science Publishers, New York-London-Paris) (1968; Zbl 0189.44201)] instead of the original paper of of C. Foiaş, quoted as [120] [see Arch. Math. 14, 341–349 (1963; Zbl 0176.43802)], where the concept was introduced and its basic properties were already obtained.

The first chapter of the monograph deals with the Kato decomposition property, which holds for a class of linear operators containing the semi-Fredholm operators. The concepts of semi-regular, essentially semi-regular and Kato type operators are introduced, all of them leading to distinguished spectra.

The second chapter is devoted to a localized version of the single-valued extension property at a point, while the third chapter describes the connections between local spectral theory and Fredholm theory.

The fourth chapter is devoted to the theory of multipliers of commutative semi-simple Banach algebras. Desiring to make the work as much self-contained as possible, the elementary theory of multipliers is included. Although some preliminary results are investigated in the non-commutative case, the author focuses his attention on commutative algebras, where the Gelfand theory is available.

The introduction of basic facts concerning the abstract Fredholm theory in arbitrary Banach algebras forms the first part of the fifth chapter. An important role in the discussion is played by the concepts of inessential ideal and the socle in a Banach algebra. We recall that the inessential ideal of the socle corresponds in the case of the Banach algebra of all bounded linear operators to the ideal of all bounded finite rank operators.

Riesz algebras, compact multipliers and Weyl multipliers are also studied in the fifth chapter. The theory of decomposable operators, introduced by C. Foiaş in 1963, is the main subject of the sixth chapter. Using various simplifications of the original definition of Foiaş, and after the presentation of their general properties, the author deals with decomposable operators belonging to several classes: right shift operators, multipliers, convolution operators, etc.

The seventh and last chapter of this monograph is devoted to the study of some perturbation classes of operators as, for instance, the classes of Fredholm operators, upper and lower semi-Fredholm operators, and various ideals of operators. Topics such as inessential operators between Banach spaces, strictly singular and strictly cosingular operators, and improjective operators are also approached in this chapter.

This 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. Unfortunately, a slight carelessness can be sometimes detected. For instance, regarding the introduction of decomposability (see the author’s comment on page 309 of his book), due to C. Foiaş, the author prefers to mention the monograph quoted as [83] [see I. Colojoara and C. Foias, “Theory of generalized spectral operators” (Mathematics and its Applications 9) (Gordon and Breach Science Publishers, New York-London-Paris) (1968; Zbl 0189.44201)] instead of the original paper of of C. Foiaş, quoted as [120] [see Arch. Math. 14, 341–349 (1963; Zbl 0176.43802)], where the concept was introduced and its basic properties were already obtained.

### MSC:

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

46H05 | General theory of topological algebras |

47A11 | Local spectral properties of linear operators |

47A53 | (Semi-) Fredholm operators; index theories |

47B40 | Spectral operators, decomposable operators, well-bounded operators, etc. |