Linear biseparating maps between vector-valued little Lipschitz function spaces

In this paper we provide a complete description of linear biseparating maps between spaces lip0(X ., E) of Banach-valued little Lipschitz functions vanishing at infinity on locally compact Hölder metric spaces X . = (X, d . X ) with 0 < . < 1. Namely, it is proved that any linear bijection T: lip0(X ., E) . lip0(Y ., 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(.(y))), where . 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, . becomes a locally Lipschitz homeomorphism and h a locally Lipschitz map from Y . 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.