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 $T^{\left\{2\right\}}\in B(Y,X)$ and $\delta T\in B(X,Y)$ with $\Vert \delta T{T}^{\left\{2\right\}}\Vert <1$. What condition on the small perturbation $\delta T$ can guarantee that the simplest possible expression $B=T^{\left\{2\right\}}(I+\delta T{T}^{\left\{2\right\}}{)}^{-1}$ 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.