The g-Drazin inverses of anti-triangular block operator matrices

Abstract

An element a in a Banach algebra A has g-Drazin inverse if there exists b∈A such that ab=ba,b=bab and a−a2b∈Aqnil. In this paper we find new explicit representations of the g-Drazin inverse of the block operator matrix (EIF0). We thereby solve a wider kind of singular differential equations posed by Campbell (1983) [2].