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Maps preserving common zeros between subspaces of vector-valued continuous functions

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Abstract

For metric spaces X and Y, normed spaces E and F, and certain subspaces A(X, E) and A(Y, F) of vector-valued continuous functions, we obtain a complete characterization of linear and bijective maps \({T:A(X,E)\rightarrow A(Y,F)}\) preserving common zeros, that is, maps satisfying the property

$$Z(f) \cap Z(g) \neq \emptyset \Longleftrightarrow Z(Tf) \cap Z(Tg) \neq \emptyset \quad\quad\quad{\rm (P)}$$

for any \({f, g \in A(X, E)}\), where \({Z(f) = \{x \in X: f(x) = 0\}}\). Moreover, we provide some examples of subspaces for which the automatic continuity of linear bijections having the property (P) is derived.

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  1. Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avenida de los Castros s/n, 39071, Santander, Spain

    Luis Dubarbie

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  1. Luis Dubarbie
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Correspondence to Luis Dubarbie.

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Research supported by the Spanish Ministry of Science and Education (MTM2006-14786) and by a predoctoral grant from the University of Cantabria and the Government of Cantabria.

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Dubarbie, L. Maps preserving common zeros between subspaces of vector-valued continuous functions. Positivity 14, 695–703 (2010). https://doi.org/10.1007/s11117-010-0046-z

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