New additive results for the generalized Drazin inverse

In this paper, we investigate additive properties of generalized Drazin inverse of two Drazin invertible linear operators in Banach spaces. Under the commutative condition of P Q = Q P , we give explicit representations of the generalized Drazin inverse ( P + Q ) d in term of P, P d , Q and Q d . We consider some applications of our results to the perturbation of the Drazin inverse and analyze a number of special cases.