Representations for the Generalized Drazin Inverse of the Sum in a Banach Algebra and Its Application for Some Operator Matrices

Abstract

We investigate additive properties of the generalized Drazin inverse in a Banach algebra A. We find explicit expressions for the generalized Drazin inverse of the sum a + b, under new conditions on a, b ∈ A. As an application we give some new representations for the generalized Drazin inverse of an operator matrix.

The g-Drazin inverse of the sum in Banach algebras

“…The explicit representation of (a + b) d is also presented. This extends [11,Theorem 4] to more general setting.…”

Expressions for the g-Drazin inverse in a Banach algebra

“…The explicit representation of (a + b) d is presented. This extends [11,Theorem 4] to more general setting.…”

“…The formula for M d is given. This extends [11,Theorem 10] to the wider case. Finally, in the last section, we present certain simpler representations of the g-Drazin inverse of the block matrix M. If BC = 0 and…”

“…The following example illustrates that Theorem 4.3 is a nontrivial generalization of [11,Theorem 10]. In this case, AB A π BD.…”

“…The g-Drazin invertibility of the sum of two elements in a Banach algebra is attractive. Many authors have studied such problems from many different views, e.g., [3,4,6,7,11,13,15,17]. In Section 2, we investigate when the sum of two g-Drazin invertible elements in a Banach algebra has g-Drazin inverse.…”