The G-Drazin Inverse of an Operator Matrix over Banach Spaces

Let A be a Banach algebra. An element a is an element of A has generalized Drazin inverse if there exists b is an element of A such that b = bab, ab = ba, a - a (2) b is an element of A (qnil) . New additive results for the generalized Drazin inverse of an operator over a Banach space are presented. we extend the main results of a paper of Shakoor, Yang and Ali from 2013 and of Wang, Huang and Chen from 2017. Appling these results to 2 x2 operator matrices we also generalize results of a paper of Deng, Cvetkovic<acute accent>-Ilic<acute accent> and Wei from 2010.