The characterizations of the stable perturbation of a closed operator by a linear operator in Banach spaces

Abstract

In this paper, we investigate the invertibility of $I_Y+δTT^+$ when $T$ is a closed operator from $X$ to $Y$ with a generalized inverse $T^+$ and $δT$ is a linear operator whose domain contains $D(T)$ and range is contained in $D(T^+)$. The characterizations of the stable perturbation $T+δT$ of $T$ by $δT$ in Banach spaces are obtained. The results extend the recent main results of Huang's in Linear Algebra and its Applications.