We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of

Additive perturbation results for the generalized Drazin inverse of Banach space operators are presented. Precisely, ifA^ddenotes the generalized Drazin inverse of a bounded linear operatorAon an arbitrary complex Banach space, then in some special cases(A + B)^dis computed in terms ofA^dandB^d. Thus, recent results of Hartwig, Wang and Wei (Linear Algebra Appl.322 (2001), 207–217) are extended to infinite dimensional settings with simplified proofs.

We define a new generalized inverse (named the gDMP inverse) for a Hilbert space operator using its generalized Drazin inverse and its Moore–Penrose inverse. Thus, we extend the DMP inverse for a square matrix to more general case. Also, we introduce two new classes of operators, g -EP and g -normal operators which include, respectively, EP operators and normal operators. A new binary relation is associated with the gDMP inverse is presented and studied. The notion of core-EP inverse for matrices is extended to generalized Drazin invertible operators on Hilbert space.