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

Published online by Cambridge University Press:  24 October 2008

Fernando Bombal
Departamento de Teoría de Funciones, Universidad Complutense de Madrid, Spain
Pilar Cembranos
Departamento de Teoría de Funciones, Universidad Complutense de Madrid, Spain


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.

Research Article
Copyright © Cambridge Philosophical Society 1985

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[1]Arens, R.. Extension of functions on fully normal spaces. Pacific J. Math. 2 (1952), 1122.CrossRefGoogle Scholar
[2]Batt, J. and Berg, E. J.. Linear bounded transformations on the space of continuous functions. J. Funct. Anal. 4 (1969), 215239.CrossRefGoogle 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), 139162.CrossRefGoogle Scholar
[5]Cembbanos, P.. On Banach spaces of vector valued continuous functions. Bull. Austral. Math. Soc. 28 (1983), 175186.CrossRefGoogle 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).CrossRefGoogle Scholar
[8]Dobrakov, I.. On representation of linear operators on C0(T, X). Czechoslovak Math. J. 21 (1971), 1330.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), 6570.CrossRefGoogle Scholar
[11]Grothendieck, A.. Sur les applications linéaires faiblement compactes d'espaces du type C(K). Canad. J. Math. 5 (1963), 129173.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).CrossRefGoogle Scholar
[14]Lindenstrauss, J. and Tzafriri, L.. Classical Banach Spaces I (Springer-Verlag, 1977).CrossRefGoogle 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), 101108.CrossRefGoogle 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), 8590.CrossRefGoogle Scholar