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

Abstract

Given a bounded operator A on a Banach space X with Drazin inverse A D 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 ) ∩ N ( A r ) = { 0 } . We show that they can be written with respect to the decomposition X = R ( A r ) ⊕ N ( A r ) as a matrix operator, B = ( B 1 B 12 B 21 B 21 B 1 −1 B 12 ) , where B 1 and B 1 2 + B 12 B 21 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 ♯ − A D ‖ and ‖ B B ♯ − A D A ‖ . We obtain a result on the continuity of the group inverse for operators on Banach spaces.