Perturbation and expression for inner inverses in Banach spaces and its applications

It is well known that if T is invertible and T-1 is its inverse, then is the inverse of for . The main purpose of this paper is to study the following problem: if T- is an inner inverse of T and , is an inner inverse of ? If the answer is no, when is an inner inverse of ? In this paper, we give a complete answer to these problems and provide some characterizations for to be an inner inverse of in Banach spaces. Utilizing these results, we investigate similar problems for the generalized inverse, {1, 3}-inverse, {1, 4}-inverse, {1, 5}-inverse, {1, 2, 3}-inverse, {1, 2, 4}-inverse, Moore–Penrose inverse, group inverse and Drazin inverse. The results obtained in this paper extend and improve many recent results in this area.