## 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:

  DOI: 10.1006/jmaa.1997.5267 · Zbl 0885.46035  DOI: 10.1007/BF01189099 · Zbl 0858.54022  Aliprantis, Positive operators (1985)  Abramovich, Linear Operators and Their Applications pp 13– (1981)  Abramovich, Dokl. Akad. Nauk USSR 248 pp 1033– (1979)  Abramovich, Inverses of disjointness preserving operators (2000)  DOI: 10.1023/A:1009830629613 · Zbl 0986.47032  DOI: 10.1090/S0002-9939-98-04318-4 · Zbl 0891.47024  Abramovich, Netherl. Acad. Wetensch. Proc. Ser. A 86 pp 265– (1983)  Jarosz, Canad. Math. Bull. 33 pp 139– (1990) · Zbl 0714.46040  Jamison, J. Operator Theory 20 pp 307– (1988)  DOI: 10.1007/BF01189890 · Zbl 0805.46049  Chan, J. Operator Theory 24 pp 383– (1990)  DOI: 10.1007/BF01246833 · Zbl 0666.46018  DOI: 10.2307/2045952 · Zbl 0645.46065  Beckenstein, Acad. Roy. Beig. Bull. Cl. Sci. 73 pp 191– (1987)  DOI: 10.1007/BF02568002 · Zbl 0827.46032  Araújo, Proceedings of the Eighth Summer Conference on Topology, Queens College, New York, June 1992 pp 296– (1994)  Vulikh, Mat. Sb. N.S. 22 pp 267– (1948)  Vulikh, Dokl. Akad. Nauk USSR 41 pp 148– (1943)  Lau, Pacific J. Math. 60 pp 229– (1975) · Zbl 0322.46042  DOI: 10.2307/1969472 · Zbl 0038.27301  DOI: 10.1006/jmaa.1996.0296 · Zbl 0936.47011
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