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Elements of rings and Banach algebras with related spectral idempotents

Part of: Basic linear algebra

Published online by Cambridge University Press:  09 April 2009

N. Castro-González  and
J. Y. Vélez-Cerrada
N. Castro-González
Affiliation:
Facultad de Informática, Universidad Politécnica de Madrid, 28660 Boadilla del Monte, Madrid, Spain, e-mail: [email protected], [email protected]
J. Y. Vélez-Cerrada
Affiliation:
Facultad de Informática, Universidad Politécnica de Madrid, 28660 Boadilla del Monte, Madrid, Spain, e-mail: [email protected], [email protected]
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Abstract

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Let aπ denote the spectral idempotent of a generalized Drazin invertible element a of a ring. We characterize elements b such that 1 − (bπ − aπ)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 ∈ (X) such that pr(Bπ) = pr(Aπ + S), where pr is the natural homomorphism of (X) onto the Calkin algebra and S ∈(X) is given.

2000 Mathematics subject classification: primary 16A32, 16A28, 15A09.

Keywords

Generalized Drazin inversespectral idempotentEP elementsperturbationFredholm operators.

MSC classification

Secondary: 15A09: Matrix inversion, generalized inverses
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 3 , June 2006 , pp. 383 - 396
Copyright
Copyright © Australian Mathematical Society 2006

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