A generalized Drazin inverse

Abstract

1. Introduction. The main theme of this paper can be described as a study of the Drazin inverse for bounded linear operators in a Banach space X when 0 is an isolated spectral point of the operator. This inverse is useful for instance in the solution of differential equations formulated in a Banach space X. Since the elements of X rarely enter into our considerations, the exposition seems to gain in clarity when the operators are regarded as elements of the Banach algebra L(X).An element a of a complex Banach algebra A is called regular (or relatively regular) if there is x e A such that axa = a. Relatively regular elements have been extensively studied in the case that A is the Banach algebra L{X) of all bounded linear operators on a complex Banach space X\ they have been shown to generalize in certain aspects invertible operators.If a is relatively regular, then it has a generalized inverse, which is an element b e A satisfying the equations aba = a and bab = b. (See [18] for a comprehensive account of generalized inverses.) A relation between a relatively regular element and its generalized inverse is reflexive in the sense that if b is a generalized inverse of a, then a is a generalized inverse of b.In 1958, Drazin [7] introduced a different kind of a generalized inverse in associative rings and semigroups that does not have the reflexivity property but commutes with the element.

Generalized Drazin invertibility of the product and sum of two elements in a Banach algebra and its applications

“…The generalized Drazin inverse in a Banach algebra was introduced in [10]. The expressions for the generalized Drazin inverse of the product and the sum were studied by many authors.…”

Weyl's Theorem, Tensor Products and Multiplication Operators Ii

“…is the commuting sum of an invertible and a nilpotent, while finally the invertibility of (8), and T ∈ G is therefore polar.…”

On Integral Representations of the Drazin Inverse in Banach Algebras

“…Let A be a complex unital Banach algebra with unit e. In [4], a generalized Drazin inverse of an element a ∈ A was defined as b ∈ A such that…”