Expressions for the g-Drazin inverse in a Banach algebra

Abstract

We explore the generalized Drazin inverse in a Banach algebra. Let $\mathcal{A}$ be a Banach algebra, and let $a,b\in {\mathcal{A}}^d$. If $ab=\lambda a^{\pi }ba{b}^{\pi }$ then $a+b\in {\mathcal{A}}^d$. The explicit representation of $(a+b{)}^d$ is also presented. As applications of our results, we present new representations for the generalized Drazin inverse of a block matrix in a Banach algebra. The main results of Liu and Qin [Representations for the generalized Drazin inverse of the sum in a Banach algebra and its application for some operator matrices, Sci. World J., {\bf 2015}, 156934.8] are extended.