Representations for the Generalized Drazin Inverse in a Banach Algebra (communicated by Fuad Kittaneh)

The Drazin inverse has important applications in matrix theory and fields such as statistics, probability, linear systems theory, differential and difference equations, Markov chains, and control theory ([1, 2, 11]). In [9], Koliha extended the Drazin invertibility in the setting of Banach algebras with applications to bounded linear operators on a Banach space. In this paper, Koliha was able to deduce a formula for the generalized Drazin inverse of a+ b when ab = ba = 0. The general question of how to express the generalized Drazin inverse of a+ b as a function of a, b, and the generalized Drazin inverses of a and b without side conditions, is very difficult and remains open. R.E. Hartwig, G.R. Wang, and Y. Wei studied in [8] the Drazin inverse of a sum of two matrices A and B when AB = 0. In the papers [3, 4, 5, 7], some new conditions under which the generalized Drazin inverse of the sum a+ b in a Banach algebra is explicitly expressed in terms of a, b, and the generalized Drazin inverses of a and b. In this paper we introduce some new conditions and we extend some known expressions for the generalized Drazin inverse of a+b, where a and b are generalized Drazin invertible in a unital Banach algebra. Throughout this paper we will denote by A a unital Banach algebra with unity 1. Let A −1 and A qnil denote the sets of all invertible and quasinilpotent elements in A , respectively. Explicitly,