The continuity and the simplest possible expression of inner inverses of linear operators in Banach space

Abstract

The main topic of this paper is the relationship between the continuity and the simplest possible expression of inner inverses. We first provide some new characterizations for the simplest possible expression to be an inner inverse of the perturbed operator. Then we obtain the equivalence between the continuity of inner inverse and the statement that the simplest possible expression is an inner inverse of $\overline{T}$. Furthermore, we prove that if $T_{n}\rightarrow T$ and the sequence of inner inverses $\{T_n^-\}$ is convergent, then $T$ is inner invertible and for any inner inverse $T^-$ of $T$, $B_n=T^{-}[I+(T_n-T)T^{-}]^{-1}=[I+T^{-}(T_n-T)]^{-1} T^{-}$ is an inner inverse of $T_n$ for all sufficiently large $n$. This is very useful and convenient in applications.