Group invertibility of the sum in rings and its applications

TL;DR

This paper investigates the conditions under which the sum of elements in a ring is group invertible, and applies these findings to block operator matrices, extending existing results in the literature.

Contribution

It introduces new additive criteria for group invertibility in rings and applies them to block operator matrices, generalizing previous results.

Findings

Derived new conditions for group invertibility of sums in rings

Established existence of group inverses for 2x2 block operator matrices

Extended known results in the literature on operator matrix invertibility

Abstract

We present new additive results for the group invertibility in a ring. Then we apply our results to block operator matrices over Banach spaces and derive the existence of group inverses of $2\times 2$ block operator matrices. These generalize many known results, e.g., Benitez, Liu and Zhu(Linear Multilinear Algebra, {\bf 59}(2011), 279--289) and Zhou, Chen and Zhu(Comm. Algebra, {\bf 48}(2020),676-690).