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

Gau, Hwa-Long; Jeang, Jyh-Shyang; Wong, Ngai-Ching

J. Aust. Math. Soc. 74, No. 1, 101-109 (2003).

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.

Reviewer: Raymond Mortini (Metz)

MSC:

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

Keywords:

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

References:

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