Expressions for the [formula omitted]-Drazin inverse of additive perturbed elements in a Banach algebra

Abstract

We study additive properties of the g -Drazin inverse in a Banach algebra A . In our development we derive a representation of the resolvent of a 2 × 2 matrix with entries in A , which is then used to find explicit expressions for the g -Drazin inverse of the sum a + b , under new conditions on a , b ∈ A . As an application of our results we obtain a representation for the Drazin inverse of a 2 × 2 complex block matrix in terms of the individual blocks, under certain conditions.