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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)C(Y,F) is called separating if f(x)g(x)=0 for every xE implies that (Tf)(y)(Tg)(y)=0 for every yY. 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 φ from Y onto X such that Tf(y)=h(y)(f(φ(y)) for every fC(X,E) and yY. It is also shown that T is bounded if and only if for every yY, h(y) is a bounded linear operator from E onto F. An example of an unbounded T is given.
47B33Composition operators
47B38Operators on function spaces (general)
46E40Spaces of vector- and operator-valued functions