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2016 Common zeros preserving maps on vector-valued function spaces and Banach modules
Maliheh Hosseini, Fereshteh Sady
Publ. Mat. 60(2): 565-582 (2016). DOI: 10.5565/PUBLMAT_60216_10


Let $X$, $Y$ be Hausdorff topological spaces, and let $E$ and $F$ be Hausdorff topological vector spaces. For certain subspaces $A(X, E)$ and $A(Y,F)$ of $C(X,E)$ and $C(Y,F)$ respectively (including the spaces of Lipschitz functions), we characterize surjections $S,T\colon A(X,E) \rightarrow A(Y,F)$, not assumed to be linear, which jointly preserve common zeros in the sense that $Z(f-f') \cap Z(g-g') \neq \emptyset$ if and only if $Z(Sf-Sf') \cap Z(Tg-Tg') \neq \emptyset$ for all $f,f',g,g'\in A(X,E)$. Here $Z(\cdot)$ denotes the zero set of a function. Using the notion of point multipliers we extend the notion of zero set for the elements of a Banach module and give a representation for surjective linear maps which jointly preserve common zeros in module case.


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Maliheh Hosseini. Fereshteh Sady. "Common zeros preserving maps on vector-valued function spaces and Banach modules." Publ. Mat. 60 (2) 565 - 582, 2016.


Received: 16 March 2015; Revised: 22 October 2015; Published: 2016
First available in Project Euclid: 11 July 2016

zbMATH: 1358.46048
MathSciNet: MR3521501
Digital Object Identifier: 10.5565/PUBLMAT_60216_10

Primary: 46J10 , 47B48
Secondary: 46J20

Keywords: Banach modules , maps preserving common zeros , Non-vanishing functions , point multipliers , vector-valued continuous function , Zero set

Rights: Copyright © 2016 Universitat Autònoma de Barcelona, Departament de Matemàtiques


Vol.60 • No. 2 • 2016
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