New characterizations of G-Drazin inverse in Banach algebra

TL;DR

This paper introduces a new characterization of the g-Drazin inverse in Banach algebras, providing necessary and sufficient conditions and extending previous results on matrix cases.

Contribution

It offers a novel characterization of the g-Drazin inverse in Banach algebras and extends existing results to certain matrix cases.

Findings

New characterization of g-Drazin inverse in Banach algebra

Necessary and sufficient conditions for existence of g-Drazin inverse

Extension of previous matrix results to Banach algebra context

Abstract

In this paper, we present a new characterization of g-Drazin inverse in a Banach algebra. We prove that an element a is a Banach algebra has g-Drazin inverse if and only if there exists $x\in A$ such that $ax=xa, a-a^2x\in A^{qnil}$. we obtain the sufficient and necessary conditions for the existence of the g-Drain inverse for certain $2 \times 2$ anti-triangular matrices over a Banach algebra. These extend the results of Koliha (Glasgow Math. J., 38(1996), 367-381), Nicholson (Comm. Algebra,27(1999), 3583-3592 and Zou et al. (Studia Scient. Math. Hungar., 54(2017,489-508).