Characterizations and Extensions of Lipschitz–α Operators

Abstract

In this work, we prove that a map F from a compact metric space K into a Banach space X over ${\Bbb F}$ is a Lipschitz–α operator if and only if for each σ in X* the map σ∘F is a Lipschitz–α 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 σ ↦ (σ ∘ 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 < α ≤ 1, we show that 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–α operator is also obtained.