J. Aust. Math. Soc. 83 (2007), no. 2, pp. 271–284.

Stable perturbation in Banach algebras

Yifeng Xue
Department of Mathematics
East China Normal University
Shanghai 200062
P.R. China
xyf63071@public9.sta.net.cn
yfxue@math.edu.cn
Received 19 March 2005; revised 22 September 2006
Communicated by A. Pryde

Abstract

Let \mathcal {A} be a unital Banach algebra. Assume that a has a generalized inverse a^+. Then \bar a=a+\delta a\in \mathcal {A} is said to be a stable perturbation of a if \bar a\mathcal {A}\cap (1-aa^+)\mathcal {A}=\{0\}. In this paper we give various conditions for stable perturbation of a generalized invertible element and show that the equation \bar a\mathcal {A}\cap (1-aa^+)\mathcal {A}=\{0\} is closely related to the gap function \mathop {\hat \delta }(\bar a\mathcal {A},a\mathcal {A}). These results will be applied to error estimates for perturbations of the Moore–Penrose inverse in C^*-algebras and the Drazin inverse in Banach algebras.

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2000 Mathematics Subject Classification: primary 46H99, 46N40; secondary 65J05
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