The Continuity and the Simplest Possible Expression of Inner Inverses of Linear Operators in Banach Space

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 conditions on the continuity of the inner inverse. Furthermore, we prove that if T-n -> T and the sequence of inner inverses [T-n(-)] g is convergent, then T is inner invertible and we can find a succinct expression of the inner inverse of T-n, which converge to any given inner inverse T-. This is very useful and convenient in applications.