Linear biseparating maps between vector-valued little Lipschitz function spaces

Abstract

In this paper we provide a complete description of linear biseparating maps between spaces lip0(X α , E) of Banach-valued little Lipschitz functions vanishing at infinity on locally compact Holder metric spaces X α = (X, d α ) with 0 < α < 1. Namely, it is proved that any linear bijection T: lip0(X α , E) → lip0(Y α , F) satisfying that |Tf(y)| F |Tg(y)| F = 0 for all y ∈ Y if and only if |f(x)| E |g(x)| E = 0 for all x ∈ X, is a weighted composition operator of the form Tf(y) = h(y)(f(φ(y))), where φ is a homeomorphism from Y onto X and h is a map from Y into the set of all linear bijections from E onto F. Moreover, T is continuous if and only if h(y) is continuous for all y ∈ Y. In this case, φ becomes a locally Lipschitz homeomorphism and h a locally Lipschitz map from Y α into the space of all continuous linear bijections from E onto F with the metric induced by the operator canonical norm. This enables us to study the automatic continuity of T and the existence of discontinuous linear biseparating maps.