Expressions for the g-Drazin inverse of additive perturbed elements in a Banach algebra

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.