Perturbation bounds for the Moore–Penrose metric generalized inverse in some Banach spaces

Abstract

Let $X,Y$ be Banach spaces, and let $T$, $\delta T:X\to Y$ be bounded linear operators. Put $\overline{T}=T+\delta T$. In this article, utilizing the gap between closed subspaces and the perturbation bounds of metric projections, we first present some error estimates of the upper bound of $\Vert {\overline{T}}^M-T^M\Vert$ in $L^p$ ($1<p<+\infty$) spaces. Then, by using the concept of strong uniqueness and modulus of convexity, we further investigate the corresponding perturbation bound $\Vert {\overline{T}}^M-T^M\Vert$ in uniformly convex Banach spaces.