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Abstract: We obtain several Banach–Stone type theorems for vector-valued functions in this paper. Let be realcompact or metric spaces, locally convex spaces, and a bijective linear map from onto . If preserves zero set containments, i.e., then is homeomorphic to , and is a weighted composition operator. The above conclusion also holds if we assume a seemingly weaker condition that preserves nonvanishing functions, i.e., These two results are special cases of the theorems in a very general setting in this paper, covering bounded continuous vector-valued functions on general completely regular spaces, and uniformly continuous vector-valued functions on metric spaces. Our results extend and generalize many recent ones. [Copyright &y& Elsevier]

For a linear isometry T :C0(X)→C0(Y) of finite corank, there is a cofinite subset Y1 of Y such that Tf&z.sfnc;Y1=h·f∘ϕ is a weighted composition operator and X is homeomorphic to a quotient space of Y1 modulo a finite subset. When X=Y, such a T is called an isometric quasi-n-shift on C0(X). In this case, the action of T can be implemented as a shift on a tree-like structure, called a T-tree, in M(X) with exactly n joints. The T-tree is total in M(X) when T is a shift. With these tools, we can analyze the structure of T. [Copyright &y& Elsevier]