Characterizations and Extensions of Lipschitz Operators

In this work, we prove that a map F from a compact metric space K into a Banach space X over F is a Lipschitz.. operator if and only if for each . in X* the map ..F is a Lipschitz.. function on K. In the case that K = [a, b], we show that a map f from [a, b] into X is a Lipschitz.1 operator if and only if it is absolutely continuous and the map . . (. . f)' is a bounded linear operator from X* into L .([a, b]). When K is a compact subset of a finite interval (a, b) and 0 < . . 1, we show that every Lipschitz.. operator f from K into X can be extended as a Lipschitz.. operator F from [a, b] into X with L . (f) . L . (F) . 31.. L . (f). A similar extension theorem for a little Lipschitz.. operator is also obtained.