Metric generalized inverse of linear operator in Banach space

The Moore-Penrose metric generalized inverse T+ of linear operator T in Banach space is systematically investigated in this paper. Unlike the case in Hilbert space, even T is a linear operator in Banach Space, the Moore-Penrose metric generalized inverse T+ is usually homogeneous and nonlinear in general. By means of the methods of geometry of Banach Space, the necessary and sufficient conditions for existence, continuity, linearity and minimum property of the Moore-Penrose metric generalized inverse T+ will be given, and some properties of T+ will be investigated in this paper.