In this paper, we prove that into isometries and disjointness preserving linear Ž . Ž . maps from C X into C Y are essentially weighted composition operators 0 0Tf s h и f ( for some continuous map and some continuous scalar-valued function h. ᮊ

LetA\ y be compact Hausdorff spaces and £, F be Banach spaces. Alinearmap T :is separating if Tf, Tg have disjoint cozeroes whenever / , g have disjoint cozeroes. We prove that a biseparating linear bijection T (that is, T and T~l are separating) is a weighted composition operator Tf = hf o

Abstract. Let X and Y be locally compact Hausdorff spaces, let E and F be Banach spaces, and let T be a linear isometry from C 0 (X, E) into C 0 (Y, F ). We provide three new answers to the Banach-Stone problem: (1) T can always be written as a generalized weighted composition operator if and only if F is strictly convex; (2) if T is onto then T can be written as a weighted composition operator in a weak sense; and (3) if T is onto and F does not contain a copy of ∞ 2 then T can be written as a weighted composition operator in the classical sense.

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