Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-19T07:21:32.423Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterization of some classes of operators on spaces of vector-valued continuous functions

Published online by Cambridge University Press:  24 October 2008

Fernando Bombal  and
Pilar Cembranos
Fernando Bombal
Affiliation:
Departamento de Teoría de Funciones, Universidad Complutense de Madrid, Spain
Pilar Cembranos
Affiliation:
Departamento de Teoría de Funciones, Universidad Complutense de Madrid, Spain
Get access Rights & Permissions [Opens in a new window]

Extract

Let K be a compact Hausdorff space and E, F Banach spaces. We denote by C(K, E) the Banach space of all continuous. E-valued functions defined on K, with the supremum norm. It is well known ([6], [7]) that every operator (= bounded linear operator) T from C(K, E) to F has a finitely additive representing measure m of bounded semi-variation, defined on the Borel σ-field Σ of K and with values in L(E, F″) (the space of all operators from E into the second dual of F), in such a way that

where the integral is considered in Dinculeanu's sense.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 1 , January 1985 , pp. 137 - 146
Copyright
Copyright © Cambridge Philosophical Society 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Arens, R.. Extension of functions on fully normal spaces. Pacific J. Math. 2 (1952), 11–22.Google Scholar
[2]Batt, J. and Berg, E. J.. Linear bounded transformations on the space of continuous functions. J. Funct. Anal. 4 (1969), 215–239.Google Scholar
[3]Bilyeu, R. G. and Lewis, P. W.. Vector measures and weakly compact operators on continuous function spaces: A survey. Conference on Measure Theory and its Applications. Proceedings of the 1980 conference at Northern Illinois University.Google Scholar
[4]Brooks, J. and Lewis, P.. Linear operators and vector measures. Trans. Amer. Math. Soc. 192 (1974), 139–162.Google Scholar
[5]Cembbanos, P.. On Banach spaces of vector valued continuous functions. Bull. Austral. Math. Soc. 28 (1983), 175–186.Google Scholar
[6]Diestel, J. and Uhl, J. J.. Vector Measures. Math. surveys, no. 15 (Amer. Math. Soc., Providence, 1977).CrossRefGoogle Scholar
[7]Dinculeanu, N.. Vector Measures (Pergamon Press, 1967).Google Scholar
[8]Dobrakov, I.. On representation of linear operators on C0(T, X). Czechoslovak Math. J. 21 (1971), 13–30.Google Scholar
[9]Dunford, N. and Schwartz, J.. Linear Operators I (Interscience, 1967).Google Scholar
[10]Ghoussoub, N. and Saab, P.. Weak compactness in spaces of Bochner integrable functions and the Radon–Nikodym property. Pacific J. Math. 110 (1984), 65–70.Google Scholar
[11]Grothendieck, A.. Sur les applications linéaires faiblement compactes d'espaces du type C(K). Canad. J. Math. 5 (1963), 129–173.Google Scholar
[12]Grothendieck, A.. Topological Vector Spaces (Gordon and Breach, 1973).Google Scholar
[13]Lacey, H. E.. The Isometric Theory of Classical Banach Spaces (Springer-Verlag, 1974).Google Scholar
[14]Lindenstrauss, J. and Tzafriri, L.. Classical Banach Spaces I (Springer-Verlag, 1977).Google Scholar
[15]Saab, P.. Weakly compact, unconditionally converging, and Dunford-Pettis operators on spaces of vector-valued continuous functions. Math. Proc. Cambridge Philos. Soc. 95 (1984), 101–108.Google Scholar
[16]Semadeni, Z.. Banach Spaces of Continuous Functions (PWN, Warsaw, 1971).Google Scholar
[17]Swartz, C.. Unconditionally converging and Dunford-Pettis operators on CX(S). Studia Math. 57 (1976), 85–90.Google Scholar