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

In this paper, we investigate the invertibility of IY+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 main results in [7,8].