Criteria for the single-valued metric generalized inverses of multi-valued linear operators in banach spaces

Let X, Y be Banach spaces and M be a linear subspace in X . Y = {{x, y}|x . X, y . Y}. We may view M as a multi-valued linear operator from X to Y by taking M(x) = {y|{x, y} . M}. In this paper, we give several criteria for a single-valued operator from Y to X to be the metric generalized inverse of the multi-valued linear operator M. The principal tool in this paper is also the generalized orthogonal decomposition theorem in Banach spaces.