Generalized Drazin invertibility of the sum of two elements in a Banach algebra

Abstract

In this paper, we study additive properties of the generalized Drazin
in\-ver\-se in a Banach algebra. We first show that $a+b\in \ab^d$ under
the condition that $a, b\in \ab^d$, $ aba^{\pi}=\lambda a^{\pi} ba b^{\pi}a^{\pi}$, and then give some
explicit expressions for the generalized Drazin inverse of the
sum $a+b$ under some weaker conditions than those used in the previous papers. Some
known results are extended.