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2 results

  • GENERALIZED INVERSES OF A SUM IN RINGS

  • Part of
  • N. CASTRO-GONZÁLEZ, C. MENDES-ARAÚJO, PEDRO PATRICIO
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 82 / Issue 1 / August 2010
    Published online by Cambridge University Press:
    22 April 2010, pp. 156-164
    Print publication:
    August 2010
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  • We study properties of the Drazin index of regular elements in a ring with a unity 1. We give expressions for generalized inverses of 1−ba in terms of generalized inverses of 1−ab. In our development we prove that the Drazin index of 1−ba is equal to the Drazin index of 1−ab.

  • Elements of rings and Banach algebras with related spectral idempotents

  • Part of
  • N. Castro-González, J. Y. Vélez-Cerrada
  • Journal:
    Journal of the Australian Mathematical Society / Volume 80 / Issue 3 / June 2006
    Published online by Cambridge University Press:
    09 April 2009, pp. 383-396
    Print publication:
    June 2006
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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.