In this paper, we investigate additive properties of the generalized Drazin inverse in a Banach algebra A. We find explicit expressions for the generalized Drazin inverse of the sum a + b, under new conditions on a, b ∈ A.

We investigate additive properties of the generalized Drazin inverse in a Banach algebra A. We find explicit expressions for the generalized Drazin inverse of the sum a + b, under new conditions on a, b ∈ A. As an application we give some new representations for the generalized Drazin inverse of an operator matrix.

We present some new representations for the generalized Drazin inverse of a block matrix in a Banach algebra under conditions weaker than those used in resent papers on the subject.

The Drazin inverse has applications in a number of areas such as control theory, Markov chains, singular differential and difference equations, and iterative methods in numerical linear algebra. The study on representations for the Drazin inverse of block matrices stems essentially from finding the general expressions for the solutions to singular systems of differential equations, and then stimulated by a problem formulated by Campbell. In 1983, Campbell (Campbell et al. (1976)) established an explicit representation for the Drazin inverse of a 2 × 2 block matrix M in terms of the blocks of the partition, where the blocks A and D are assumed to be square matrices. Special cases of the problems have been studied. In 2009, Chunyuan Deng and Yimin Wei found an explicit representation for the Drazin inverse of an anti-triangular matrix M, where A and BC are generalized Drazin invertible, if A π AB=0 and BC (I-A π)=0. Afterwards, several authors have investigated this problem under some limited conditions on the blocks of M. In particular, a representation of the Drazin inverse of M, denoted by M d. In this paper, we consider the Drazin inverse of a sum of two matrices and we derive additive formulas under the conditions of ABA π =0 and BA π =0 respectively. Precisely, for a block matrix M, we give a new representation of M d under some conditions that AB=0 and DCA π =0. Moreover, some particular cases of this result related to the Drazin inverse of block matrices are also considered.

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.