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Zbl 1052.47017
Gau, Hwa-Long; Jeang, Jyh-Shyang; Wong, Ngai-Ching
Biseparating linear maps between continuous vector-valued function spaces.
(English)
[J] J. Aust. Math. Soc. 74, No. 1, 101-109 (2003). ISSN 1446-7887; ISSN 1446-8107/e

The object of the paper under review is to present some vector-valued Banach-Stone theorems. Let $X$, $Y$ be compact Hausdorff spaces, $E$ and $F$ two Banach spaces and $C(X,E)$ the Banach space of all continuous $E$-valued functions defined on $X$. A linear map $T: C(X,E) \to C(Y,F)$ is called separating if $\Vert f(x)\Vert \ \Vert g(x)\Vert =0$ for every $x\in E$ implies that $\Vert (Tf)(y)\Vert \ \Vert (Tg)(y)\Vert =0$ for every $y\in Y$. The authors show that every biseparating linear bijection $T$ (that is, a $T$ for which $T$ and $T^{-1}$ are separating) is a weighted composition operator". This means that there exists a function $h$ of $Y$ into the set of all bijective linear maps of $E$ onto $F$ and a homeomorphism $\phi$ from $Y$ onto $X$ such that $Tf(y)=h(y) (f(\phi(y))$ for every $f\in C(X,E)$ and $y\in Y$. It is also shown that $T$ is bounded if and only if for every $y\in Y$, $h(y)$ is a bounded linear operator from $E$ onto $F$. An example of an unbounded $T$ is given.
[Raymond Mortini (Metz)]
MSC 2000:
*47B33 Composition operators
47B38 Operators on function spaces
46E40 Spaces of vector-valued functions

Keywords: weighted composition operator; biseparating maps; vector-valued function spaces; Banach-Stone theorems

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