Expressions for the g-Drazin inverse in a Banach algebra

TL;DR

This paper investigates the properties and explicit formulas for the generalized Drazin inverse in a Banach algebra, including sum and block matrix representations, extending previous results in the field.

Contribution

It provides new explicit representations for the generalized Drazin inverse of sums and block matrices in Banach algebras, extending prior work by Liu and Qin.

Findings

Derived conditions for the sum of two elements to be Drazin invertible.

Presented explicit formulas for the Drazin inverse of sums in Banach algebras.

Extended previous representations to broader classes of operator matrices.

Abstract

We explore the generalized Drazin inverse in a Banach algebra. Let $\mathcal{A}$ be a Banach algebra, and let $a,b\in \mathcal{A}^{d}$. If $ab=\lambda a^{\pi}bab^{\pi}$ then $a+b\in \mathcal{A}^{d}$. The explicit representation of $(a+b)^d$ is also presented. As applications of our results, we present new representations for the generalized Drazin inverse of a block matrix in a Banach algebra. The main results of Liu and Qin [Representations for the generalized Drazin inverse of the sum in a Banach algebra and its application for some operator matrices, Sci. World J., {\bf 2015}, 156934.8] are extended.