Biseparating linear maps between continuous vector-valued function spaces

Abstract. Let X, Y be compact Hausdorff spaces and E, F be Banach spaces. A linear map T: C(X,E) → C(Y, F) is separating if Tf, Tg have disjoint cozeroes whenever f, g have disjoint cozeroes. We prove that a biseparating linear bijection T (i.e., T and T−1 are separating) is a weighted composition operator Tf = h·f ◦ϕ, where ϕ is a homeomorphism from Y onto X; and T is bounded if and only if h(y) is a bounded operator from E onto F for all y in Y. 1