Representation of the g-Drazin inverse in a Banach algebra

TL;DR

This paper investigates the properties of the g-Drazin inverse within complex Banach algebras, providing new formulas for the inverse of sums of elements under specific conditions, extending previous results and applying to block operator matrices.

Contribution

It introduces a new formula for the g-Drazin inverse of a sum of elements in a Banach algebra under a specific product condition, extending prior work.

Findings

Derived a new formula for the g-Drazin inverse of a sum of elements.

Extended Mosic's main results to a more general setting.

Applied the findings to block operator matrices.

Abstract

Let $\mathcal{A}$ be a complex Banach algebra. An element $a\in \mathcal{A}$ has g-Drazin inverse if there exists $b\in \mathcal{A}$ such that $$b=bab, ab=ba, a-a^2b\in \mathcal{A}^{qnil}.$$ Let $a,b\in \mathcal{A}$ have g-Drazin inverses. If $$ab = \lambda a^\pi b^\pi b a b^\pi,$$ we prove that $a+b$ has g-Drazin inverse and $$(a+b)^d = b^\pi a^d + b^da^\pi + \sum\limits_{n=0}^\infty (b^d)^{n+1} a^n a^\pi + \sum\limits_{n=0}^\infty b^\pi (a+b)^n b(a^d)^{n+2}.$$ The main results of Mosic (Bull. Malays. Sci. Soc., {\bf 40}(2017), 1465--1478) is thereby extended to the general case. Applications to block operator matrices are given.