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.


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