Skip to main content

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

  • Fernando Bombal (a1) and Pilar Cembranos (a1)

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.

Hide All
[1]Arens R.. Extension of functions on fully normal spaces. Pacific J. Math. 2 (1952), 1122.
[2]Batt J. and Berg E. J.. Linear bounded transformations on the space of continuous functions. J. Funct. Anal. 4 (1969), 215239.
[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.
[4]Brooks J. and Lewis P.. Linear operators and vector measures. Trans. Amer. Math. Soc. 192 (1974), 139162.
[5]Cembbanos P.. On Banach spaces of vector valued continuous functions. Bull. Austral. Math. Soc. 28 (1983), 175186.
[6]Diestel J. and Uhl J. J.. Vector Measures. Math. surveys, no. 15 (Amer. Math. Soc., Providence, 1977).
[7]Dinculeanu N.. Vector Measures (Pergamon Press, 1967).
[8]Dobrakov I.. On representation of linear operators on C0(T, X). Czechoslovak Math. J. 21 (1971), 1330.
[9]Dunford N. and Schwartz J.. Linear Operators I (Interscience, 1967).
[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.
[11]Grothendieck A.. Sur les applications linéaires faiblement compactes d'espaces du type C(K). Canad. J. Math. 5 (1963), 129173.
[12]Grothendieck A.. Topological Vector Spaces (Gordon and Breach, 1973).
[13]Lacey H. E.. The Isometric Theory of Classical Banach Spaces (Springer-Verlag, 1974).
[14]Lindenstrauss J. and Tzafriri L.. Classical Banach Spaces I (Springer-Verlag, 1977).
[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.
[16]Semadeni Z.. Banach Spaces of Continuous Functions (PWN, Warsaw, 1971).
[17]Swartz C.. Unconditionally converging and Dunford-Pettis operators on CX(S). Studia Math. 57 (1976), 8590.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 5 *
Loading metrics...

Abstract views

Total abstract views: 78 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 20th November 2017. This data will be updated every 24 hours.