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 inverse in a Banach algebra. We first show that a + b ? Ad under the condition that a, b ? Ad, aba? = ?a?bab?a?, 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.