## Biseparating linear maps between continuous vector-valued function spaces.(English)Zbl 1052.47017

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 $$\| f(x)\| \;\| g(x)\| =0$$ for every $$x\in E$$ implies that $$\| (Tf)(y)\| \;\| (Tg)(y)\| =0$$ 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.

### MSC:

 47B33 Linear composition operators 47B38 Linear operators on function spaces (general) 46E40 Spaces of vector- and operator-valued functions
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### References:

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