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 +AD(B−A) is invertible andR(B)∩N(Ar ) = {0}.We show that they can be written with respect to the decomposition X =R(Ar )⊕N(Ar ) as a matrix operator, B = B1 B12 B21 B21B −1 1 B12 , where B1 and B2 1 +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 B − AD and BB − ADA . We obtain a result on the continuity of the group inverse for operators on Banach spaces. © 2007 Elsevier Inc. All rights reserved