  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).
