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 lip_0(X^α,E) of Banach-valued little Lipschitz functions vanishing at infinity on locally compact Hölder metric spaces X^α=(X,d^α) with 0 < α < 1. Namely, it is proved that any linear bijection T : lip_0(X^α,E) → lip_0(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.