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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Abstract. We characterize compact and completely continuous disjointness preserving linear operators on vector-valued continuous functions as follows: a disjointness preserving operator T : C 0 (X, E) → C 0 (Y, F ) is compact (resp. completely continuous) if and only if

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We study disjointness preserving (quasi-)n-shift operators on C 0 (X), where X is locally compact and Hausdorff. When C 0 (X) admits a quasi-n-shift T , there is a countable subset of X ∞ = X ∪ {∞} equipped with a tree-like structure, called ϕ-tree, with exactly n joints such that the action of T on C 0 (X) can be implemented as a shift on the ϕ-tree. If T is an n-shift, then the ϕ-tree is dense in X and thus X is separable. By analyzing the structure of the ϕ-tree, we show that every (quasi-)n-shift on c 0 can always be written as a product of n (quasi-)1-shifts. Although it is not the case for general C 0 (X) as shown by our counter examples, we can do so after dilation.