Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T14:55:16.073Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 operators 47B38: Operators on function spaces (general)
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

References

[1]Abramovich, Y. A., ‘Multiplicative representation of operators preserving disjointness’, Netherl. Acad. Wetensch. Proc. Ser. A 86 (1983), 265–279.Google Scholar
[2]Abramovich, Y. A. and Kitover, A. K., ‘A solution to a problem on invertible disjointness preserving operators’, Proc. Amer. Math. Soc. 126 (1998), 1501–1505.CrossRefGoogle Scholar
[3]Abramovich, Y. A. and Kitover, A. K., ‘A characterization of operators preserving disjointness in terms of their inverse’, Positivity 4 (2000), 205–212.CrossRefGoogle Scholar
[4]Abramovich, Y. A. and Kitover, A. K., Inverses of disjointness preserving operators, Mem. Amer. Math. Soc. 143 (Amer. Math. Soc., Providence, 2000).CrossRefGoogle Scholar
[5]Abramovich, Y. A., Veksler, A. I. and Koldunov, A. V., ‘Operators preserving disjointness’, Dokl. Akad. Nauk USSR 248 (1979), 1033–1036.Google Scholar
[6]Abramovich, Y. A., Veksler, A. I. and Koldunov, A. V., ‘Operators preserving disjointness, their continuity and multiplicative representation’. in: Linear Operators and Their Applications, Sb. Nauchn. Trudov (Leningrad, 1981) pp. 13–34.Google Scholar
[7]Aliprantis, C. D. and Burkinshaw, O., Positive operators (Academic Press, Orlando, 1985).Google Scholar
[8]Araújo, J., ‘Realcompactness and Banach-Stone theorems’, preprint http://xxx.lanl.gov/pdf/math/0010292.Google Scholar
[9]Araújo, J., ‘Realcompactness and spaces of vector-valued functions’, preprint http://xxx.lanl.gov/pdf/math/0010261.Google Scholar
[10]Araújo, J., Beckenstein, E. and Narici, L., ‘Biseparating maps and ring of continuous functions’, in: Proceedings of the Eighth Summer Conference on Topology, Queens College, New York, June 1992, Ann. New York Acad. Sci. 728 (New York Acad. Sci., New York, 1994) pp. 296–303.Google Scholar
[11]Beckenstein, E., Araújo, J. and Narici, L., ‘When is a separating map biseparating?’, Arch. Math. (Basel) 67 (1996), 395–407.Google Scholar
[12]Beckenstein, E., Hernández, S. and Narici, L., ‘Banach-Stone theorems and separating maps’, Manuscripta Math. 86 (1995), 409–416.Google Scholar
[13]Beckenstein, E. and Narici, L., ‘Automatic continuity of certain linear isomorphism’, Acad. Roy. Beig. Bull. Cl. Sci. (5) 73 (1987), 191–200.Google Scholar
[14]Beckenstein, E. and Narici, L., ‘A nonarchimedean Stone-Banach theorem’, Proc. Amer Math. Soc. 100 (1987), 242–246.Google Scholar
[15]Beckenstein, E. and Narici, L., ‘Automatic continuity of linear maps of spaces of continuous functions’, Manuscripta Math. 62 (1988), 257–275.CrossRefGoogle Scholar
[16]Chan, J. T., ‘Operators with the disjoint support property’, J. Operator Theory 24 (1990), 383–391.Google Scholar
[17]Font, J. J. and Hernández, S., ‘On separating maps between locally compact spaces’, Arch. Math. (Basel) 63 (1994), 158–165.CrossRefGoogle Scholar
[18]Jamison, J. E. and Rajagopalan, M., ‘Weighted composition operator on C(X, E)’, J. Operator Theory 20 (1988), 307–317.Google Scholar
[19]Jarosz, K., ‘Automatic continuity of separating linear isomorphisms’, Canad. Math. Bull. 33 (1990), 139–144.CrossRefGoogle Scholar
[20]Jeang, J. S. and Wong, N. C., ‘On the Banach-Stone problem’, preprint.Google Scholar
[21]Jeang, J. S. and Wong, N. C., ‘Weighted composition operators of C 0(X)'s’, J. Math. Anal. Appl. 201 (1996), 981–993.CrossRefGoogle Scholar
[22]Jeang, J. S. and Wong, N. C., ‘Into isometries of C 0(X, E)'s’, J. Math. Anal. Appl. 207 (1997), 286–290.CrossRefGoogle Scholar
[23]Jerison, M., ‘The space of bounded maps into a Banach space’, Ann. of Math. 52 (1950), 309–327.CrossRefGoogle Scholar
[24]Lau, K. S., ‘A representation theorem for isometries of C(X, E)’, Pacific J. Math. 60 (1975), 229–233.Google Scholar
[25]Vulikh, B. Z., ‘On linear multiplicative operations’, Dokl. Akad. Nauk USSR 41 (1943), 148–151.Google Scholar
[26]Vulikh, B. Z., ‘The product in linear partially-ordered spaces and its application to the theory of operations II’, (Russian) Mat. Sb. N.S. 22 (1948), 267–317.Google Scholar