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

Maps preserving common zeros between subspaces of vector-valued continuous functions 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

, Volume 14 (4) – Feb 9, 2010
9 pages

/lp/springer_journal/maps-preserving-common-zeros-between-subspaces-of-vector-valued-Lf68c100ct
Publisher
Springer Journals
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-010-0046-z
Publisher site
See Article on Publisher Site

### 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.

### Journal

PositivitySpringer Journals

Published: Feb 9, 2010

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