The Best Bounded Quasi-linear Generalized Inverse of the Linear Operator in Bnanch Spaces

Let X,Y be Banach spaces, T be a linear operator from X to Y. The bounded quasi-linear generalized inverseT(h) (which, itself is an extension of matrix generalized inverse, the single-valued metric generalized inverseT(M), and the continuous linear projector generalized inverseT(+)) of the linear operator T in arbitrary Banach spaces is investigated, If X and Y are reflexive, we prove that the set of all bounded quasi-linear generalized inverse of T is not empty. In Banach space of all bounded homogeneous operators, the best bounded quasi-linear generalized inverseT(h) of T is just the Moore-Penrose metric generalized inverseT(M). These results indeed extend and improve the corresponding work done by P. Liu and Y. W. Wang in 2007 from reflexive and strictly convex Banach spaces to the general Banach spaces.