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 $$b=bab,ab=ba,a-a^{2}b\in {\mathcal{A}}^{qnil}.$$ Let $a,b\in \mathcal{A}$ have g-Drazin inverses. If $$ab=\lambda a^{\pi }{b}^{\pi }ba{b}^{\pi },$$ we prove that $a+b$ has g-Drazin inverse and $$(a+b{)}^d=b^{\pi }{a}^d+b^d{a}^{\pi }+\sum _{n=0}^{\infty }(b^d{)}^{n+1}{a}^n{a}^{\pi }+\sum _{n=0}^{\infty }{b}^{\pi }(a+b{)}^{n}b(a^d{)}^{n+2}.$$ 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.