Elements of rings and Banach algebras with related spectral idempotents

J. Aust. Math. Soc. 80 (2006), 383-396

Elements of rings and Banach algebras with related spectral idempotents

N. Castro-González
Facultad de Informática
Universidad Politécnica de Madrid
28660 Boadilla del Monte
Madrid
Spain
nieves@fi.upm.es

and

J. Y. Vélez-Cerrada
Facultad de Informática
Universidad Politécnica de Madrid
28660 Boadilla del Monte
Madrid
Spain
jyvelezc@hotmail.com

Abstract

Let $a^{\pi}$ denote the spectral idempotent of a generalized Drazin invertible element

of a ring. We characterize elements

such that $1-(b^{\pi}-a^{\pi})^2$ 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 $A, B\in \mathcal{B}(X)$ such that $\operatorname{pr}(B^{\pi})=\operatorname{pr}(A^{\pi}+S)$ , where pr is the natural homomorphism of $\mathcal{B}(X)$ onto the Calkin algebra and $S\in \mathcal{B}(X)$ is given.

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