@article {Xue2007,
author="Yifeng Xue",
title={Stable perturbation in Banach algebras},
journal="J. Aust. Math. Soc.",
fjournal={Journal of the Australian Mathematical Society},
volume="83",
year="2007",
number="2",
pages="271--284",
issn="1446-7887",
coden="JAUMA2",
language="English",
date="Received 19 March 2005; revised 22 September 2006 Communicated by A. Pryde",
classmath="primary 46H99, 46N40; secondary 65J05",
publisher={AMPAI, Australian Mathematical Society},
keywords={},
url="http://www.austms.org.au/Journal+of+the+Australian+Mathematical+Society/V83P2/832-n130-Xue/index.html",
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. }
}