Let a" denote the spectral idempotent of a generalized Drazin invertible element a of a ring. We characterize elements b such that 1 -(b n -a") 2 is invertible. We also apply this result in rings with involution to obtain a characterization of the perturbation of EP elements. In Banach algebras we obtain a characterization in terms of matrix representations and derive error bounds for the perturbation of the Drazin inverse. This work extends recent results for matrices given by the same authors to the setting of rings and Banach algebras. Finally, we characterize generalized Drazin invertible operators /t, B e S8(X) such that pr(B") = pr(A" + 5), where pr is the natural homomorphism of 38(X) onto the Calkin algebra and 5 e SB(X) is given.2000 Mathematics subject classification: primary 16A32, 16A28, 15A09.

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, where B 1 and B 2 1 + 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 BB − A D A . We obtain a result on the continuity of the group inverse for operators on Banach spaces.

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