In this work,we prove that a map F from a compact metric space K into a Banach spaceX over F is a Lipschitz-α operator if and only if for each σ in X~* the map σο F is a Lipschitz-α functionon K.In the case that K =[a,b],we show that a map f from[a,b]into X is a Lipschitz-1 operatorif and only if it is absolutely continuous and the map σ(σο f)' is a bounded linear operator fromX~* into L~∞([a,b]).When K is a compact subset of a finite interval(a,b)and 0<α≤1,we showthat every Lipschitz-α operator f from K into X can be extended as a Lipschitz-α operator F from[a,b]into X with L_α(f)≤L_α(F)≤3^(1-α)L_α(f).A similar extension theorem for a little Lipschitz-αoperator is also obtained.
