The Representation of the Set-valued Metric Generalized Inverse of Linear Operator in Banach Space

Without the assumption that Banach space Y is reflexive and T is a densely defined linear operator with closed range from Banach space X to Y, by means of geometry of Banach space, it is proved that the metric generalized inverse of linear operator has closed convex range set-valued mapping. Existence, uniqueness and the equivalent representation of the metric generalized inverse are established. Parts of our results have extended and improved the corresponding results obtained recently by Y. W. Wang and S. R. Pan, H. Hudzik, Y. W. Wang and W. J. Zheng, under the assumption that Banach space Y is reflexive and T is a densely defined linear operator with closed range from X to Y.