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

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

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.

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[1] R. Arens . Extension of functions on fully normal spaces. Pacific J. Math. 2 (1952), 1122.

[2] J. Batt and E. J. Berg . Linear bounded transformations on the space of continuous functions. J. Funct. Anal. 4 (1969), 215239.

[4] J. Brooks and P. Lewis . Linear operators and vector measures. Trans. Amer. Math. Soc. 192 (1974), 139162.

[6] J. Diestel and J. J. Uhl . Vector Measures. Math. surveys, no. 15 (Amer. Math. Soc., Providence, 1977).

[10] N. Ghoussoub and P. Saab . Weak compactness in spaces of Bochner integrable functions and the Radon–Nikodym property. Pacific J. Math. 110 (1984), 6570.

[13] H. E. Lacey . The Isometric Theory of Classical Banach Spaces (Springer-Verlag, 1974).

[14] J. Lindenstrauss and L. Tzafriri . Classical Banach Spaces I (Springer-Verlag, 1977).

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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