Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2025-01-03T18:16:46.310Z Has data issue: false hasContentIssue false
  • English
  • Français

Operators on locally convex spaces of vector-valued continuous functions

Published online by Cambridge University Press:  17 April 2009

A. García López
A. García López
Affiliation:
Departamento de Teoria de Funciones, Facultad de Ciencias Matematicas, Universidad Complutense de Madrid, 28040 Madrid, Spain.
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 E and F be locally convex spaces and let K be a compact Hausdorff space. C(K,E) is the space of all E-valued continuous functions defined on K, endowed with the uniform topology.

Starting from the well-known fact that every linear continuous operator T from C(K,E) to F can be represented by an integral with respect to an operator-valued measure, we study, in this paper, some relationships between these operators and the properties of their representing measures. We give special treatment to the unconditionally converging operators.

As a consequence we characterise the spaces E for which an operator T defined on C(K,E) is unconditionally converging if and only if (Tfn) tends to zero for every bounded and converging pointwise to zero sequence (fn) in C(K,E).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 36 , Issue 2 , October 1987 , pp. 267 - 278
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Batt, J. and Berg, E.J., “Linear bounded transformations on the space of continuous functions”, J. Funct. Anal. 4 (1969), 215–239.CrossRefGoogle Scholar
[2]Bombal, F. and Cembranos, P., “Characterization of some classes of operators on spaces of vector-valued continuous functions”, Math. Proc. Cambridge Philos. Soc. 97 (1985), 137–146.CrossRefGoogle Scholar
[3]Brooks, J.K. and Lewis, P.W., “Linear operators and vector measures”, Trans. Amer. Math. Soc. 192 (1974), 139–163.CrossRefGoogle Scholar
[4]Diestel, J. and Uhl, J.J., Vector measures, (Math. Surveys 15 A.M.S. Providence R.I. 1977).CrossRefGoogle Scholar
[5]Dobrakov, I., “On representation of linear operators on C∘(T,X)”, Czechoslovak Math. J. 21 (1971), 13–30.CrossRefGoogle Scholar
[6]Dunford, N. and Schwartz, J.T., Linear operators, part I (Interscience, New York, London 1958).Google Scholar
[7]Grothendieck, A., “Sur les applications linéaires faiblement compactes d'espaces du tipe C(K), Canad. J. Math. 5 (1953), 129–173.CrossRefGoogle Scholar
[8]Jarchow, H., Locally convex spaces, (Teubner, Stuttgart 1981).CrossRefGoogle Scholar
[9]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.CrossRefGoogle Scholar
[10]Semadeni, Z., Banach spaces of continuous functions, (PWN-Polish Scientific Publishers, Warszawa 1971).Google Scholar
[11]Shuchat, A.H., “Integral representation theorems in topological vector spaces”, Trans. Amer. Math. Soc. 172 (1972), 373–397.CrossRefGoogle Scholar
[12]Swong, K., “A representation theory of continuous linear maps”, Math. Ann. 155 (1964) 270–291.CrossRefGoogle Scholar