Characterizations and extensions of Lipschitz-α 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-alpha operator if and only if for each sigma in X-* the map sigma circle F is a Lipschitz-alpha 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 sigma -> (sigma circle f)' is a bounded linear operator from X-* into L-infinity([a,b]). When K is a compact subset of a finite interval (a, b) and 0 < alpha <= 1, we show that every Lipschitz-alpha operator f from K into X can be extended as a Lipschitz-alpha operator F from [a,b] into X with L-o(f) <= L-o(F) <= 3(1-alpha)L(alpha)(f). A similar extension theorem for a little Lipschitz-alpha operator is also obtained.