Group invertibility of the sum in rings and its applications

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).