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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 $C\left(X,E\right)$ and $C\left(Y,F\right)$ the corresponding spaces of continuous functions and by $A\left(X,E\right)$ and $A\left(Y,F\right)$ subspaces of $C\left(X,E\right)$ and $C\left(Y,F\right)$, respectively. The zero set of a function $f\in C\left(X,E\right)$ is defined as $Z\left(f\right)=\left\{x\in X:f\left(x\right)=0\right\}$. The paper under review provides a complete characterization of linear and bijective maps $T:A\left(X,E\right)\to A\left(Y,F\right)$ such that $Z\left(f\right)\cap Z\left(g\right)\ne \varnothing$ if and only if $Z\left(fg\right)\ne \varnothing$ for any $f,g\in A\left(X,E\right)·$ An application to automatic continuity is included.
##### MSC:
 47B38 Operators on function spaces (general) 46E40 Spaces of vector- and operator-valued functions 46H40 Automatic continuity 47B33 Composition operators
##### References:
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