The group inverse of the sum in a Banach algebra

TL;DR

This paper establishes new conditions for the sum of two group invertible elements in a Banach algebra to also be group invertible and applies these findings to block operator matrices.

Contribution

It provides novel necessary and sufficient conditions for the group invertibility of sums in Banach algebras and extends these results to operator block matrices.

Findings

Derived new criteria for group invertibility of sums in Banach algebras

Obtained group inverses for specific block operator matrices

Extended previous results to broader classes of operators

Abstract

In this paper, we present new necessary and sufficient conditions under which the sum of two group invertible elements in a Banach algebra has group inverse. We then apply these results to block operator matrices over Banach spaces. The group inverses of certain operator block matrices are thereby obtained. Additionally, this paper extends the results obtained in \cite{L}.