On the perturbation of the group generalized inverse for a class of bounded operators in Banach spaces

Given a bounded operator A on a Banach space X with Drazin inverse AD and index r, we study the class of group invertible bounded operators B such that I + A(D)(B - A) is invertible and R(B) boolean AND N(A(r)) = {0}. We show that they can be written with respect to the decomposition X = R(A(r))circle plus N(A(r)) as a matrix operator, B = (B-1 B-12 B-21 B21B1-1B12), where B-1 and B-1(2) + B12B21 are invertible. Several characterizations of the perturbed operators are established, extending matrix results. We analyze the perturbation of the Drazin inverse and we provide explicit upper bounds of parallel to B-# - A(D)parallel to and parallel to BB# - A(D)A parallel to. We obtain a result on the continuity of the group inverse for operators on Banach spaces. (C) 2007 Elsevier Inc. All rights reserved.