Annals of Functional Analysis

On the perturbation of outer inverses of linear operators in Banach spaces

Abstract

The main concern of this article is the perturbation problem for outer inverses of linear bounded operators in Banach spaces. We consider the following perturbed problem. Let $T\in B(X,Y)$ with an outer inverse $T^{\{2\}}\in B(Y,X)$ and $\delta T\in B(X,Y)$ with $\Vert \delta TT^{\{2\}}\Vert \lt 1$. What condition on the small perturbation $\delta T$ can guarantee that the simplest possible expression $B=T^{\{2\}}(I+\delta TT^{\{2\}})^{-1}$ is a generalized inverse, Moore–Penrose inverse, group inverse, or Drazin inverse of $T+\delta T$? In this article, we give a complete solution to this problem. Since the generalized inverse, Moore–Penrose inverse, group inverse, and Drazin inverse are outer inverses, our results extend and improve many previous results in this area.

Article information

Source
Ann. Funct. Anal. (2018), 10 pages.

Dates
Accepted: 10 July 2017
First available in Project Euclid: 11 December 2017

https://projecteuclid.org/euclid.afa/1512982817

Digital Object Identifier
doi:10.1215/20088752-2017-0041

Subjects
Secondary: 47A58: Operator approximation theory

Citation

Zhu, Lanping; Pan, Weiwei; Huang, Qianglian; Yang, Shi. On the perturbation of outer inverses of linear operators in Banach spaces. Ann. Funct. Anal., advance publication, 11 December 2017. doi:10.1215/20088752-2017-0041. https://projecteuclid.org/euclid.afa/1512982817

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