Perturbation of the Moore.Penrose metric generalized inverse in reflexive strictly convex Banach spaces

Let X, Y be reflexive strictly convex Banach spaces, let T, .T: X . Y be bounded linear operators with closed range R(T). Put T.=T+.T. In this paper, by using the concept of quasiadditivity and the so called generalized Neumman lemma, we will give some error estimates of the bounds of ..T.M... By using a relation between the concepts of the reduced minimum module and the gap of two subspaces, some new existence characterization of the Moore.Penrose metric generalized inverse T.M of the perturbed operator T. will be also given.