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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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