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Biseparating linear maps between continuous vector-valued function spaces

Part of:Special classes of linear operators

Published online by Cambridge University Press:  09 April 2009

Haw-Long Gau,
Jyh-Shyang Jeang and
Nagi-Ching Wong
Haw-Long Gau
Affiliation:
Department of Mathematics National Central UniversityChung-LiTaiwan320 R.O.C. e-mail: hlgau@math.ncu.edu.tw
Jyh-Shyang Jeang
Affiliation:
Department of Applied Mathematics National Sun Yat-sen UniversityKaohsiung Taiwan 804 R.O.C. e-mail: jeangjs@math.nsysu.edu.tw wong@math.nsysu.edu.tw
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Abstract

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Let X, Y be compact Hausdorff spaces and E, F be Banach spaces. A linear map T: C(X, E) → C(Y, F) is separating if Tf, Tg have disjoint cozeroes whenever f, g have disjoint cozeroes. We prove that a biseparating linear bijection T (that is, T and T-1 are separating) is a weighted composition operator Tf = h · f o ϕ. Here, h is a function from Y into the set of invertible linear operators from E onto F, and ϕ, is a homeomorphism from Y onto X. We also show that T is bounded if and only if h(y) is a bounded operator from E onto F for all y in Y. In this case, h is continuous with respect to the strong operator topology.

MSC classification

Secondary:47B33: Composition operators47B38: Operators on function spaces (general)

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 74 , Issue 1 , February 2003, pp. 101 - 110
Copyright
Copyright © Australian Mathematical Society 2003

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