The g-Drazin invertibility in a Banach algebra

TL;DR

This paper characterizes when certain anti-triangular matrices over Banach algebras have g-Drazin inverses, introduces new additive results, and applies these findings to operator matrices, generalizing existing theorems.

Contribution

It provides necessary and sufficient conditions for g-Drazin invertibility of anti-triangular matrices and extends known results to broader classes of operator matrices.

Findings

Established conditions for g-Drazin invertibility of anti-triangular matrices.

Derived new additive properties for g-Drazin inverse.

Generalized several existing theorems on operator matrices.

Abstract

We present necessary and sufficient conditions under which the anti-triangular matrix $\left( \begin{array}{cc} a&b 1&0 \end{array} \right)$ over a Banach algebra has g-Drazin inverse. New additive results for g-Drazin inverse are obtained. Then we apply our results to $2\times 2$ operator matrices and generalize many known results, e.g.,~\cite[Theorem 2.2]{D}, ~\cite[Theorem 2.1]{YL} and \cite[Theorem 4.1]{Y}.