Representation of the g-Drazin inverse in a Banach algebra

Abstract

Let $\mathcal{A}$ be a complex Banach algebra. An element $a\in \mathcal{A}$ has g-Drazin inverse if there exists $b\in \mathcal{A}$ such that $$b=bab, ab=ba, a-a^2b\in \mathcal{A}^{qnil}.$$ Let $a,b\in \mathcal{A}$ have g-Drazin inverses. If $$ab = \lambda a^\pi b^\pi b a b^\pi,$$ we prove that $a+b$ has g-Drazin inverse and $$(a+b)^d = b^\pi a^d + b^da^\pi + \sum\limits_{n=0}^\infty (b^d)^{n+1} a^n a^\pi + \sum\limits_{n=0}^\infty b^\pi (a+b)^n b(a^d)^{n+2}.$$ The main results of Mosic (Bull. Malays. Sci. Soc., {\bf 40}(2017), 1465--1478) is thereby extended to the general case. Applications to block operator matrices are given.