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Maps preserving common zeros between subspaces of vector-valued continuous functions. (English) Zbl 1220.47042
Suppose that \(X\) and \(Y\) are metric spaces and \(E\) and \(F\) are normed vector spaces. Denote by \(C(X,E)\) and \(C(Y,F)\) the corresponding spaces of continuous functions and by \(A(X,E)\) and \(A(Y,F)\) subspaces of \(C(X,E)\) and \(C(Y,F)\), respectively. The zero set of a function \(f\in C(X,E)\) is defined as \(Z(f)=\left\{ x\in X:f(x)=0\right\}\). The paper under review provides a complete characterization of linear and bijective maps \(T:A(X,E)\rightarrow A(Y,F)\) such that \(Z(f)\cap Z(g)\neq \emptyset\) if and only if \(Z(fg)\neq\emptyset\) for any \(f,g\in A(X,E).\) An application to automatic continuity is included.

MSC:
47B38 Linear operators on function spaces (general)
46E40 Spaces of vector- and operator-valued functions
46H40 Automatic continuity
47B33 Linear composition operators
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