Integral representation for a class of vector valued operators

Author:
Lakhdar Meziani

Journal:
Proc. Amer. Math. Soc. **130** (2002), 2067-2077

MSC (2000):
Primary 28C05; Secondary 46G10

DOI:
https://doi.org/10.1090/S0002-9939-02-06336-0

Published electronically:
January 17, 2002

MathSciNet review:
1896043

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a compact space and let , be a (real, for simplicity) Banach space. We consider the space of all continuous -valued functions on , with the supremum norm .

We prove in this paper a Bochner integral representation theorem for bounded linear operators

which satisfy the following condition:

where is the conjugate space of . In the particular case where , this condition is obviously satisfied by every bounded linear operator

and the result reduces to the classical Riesz representation theorem.

If the dimension of is greater than , we show by a simple example that not every bounded linear admits an integral representation of the type above, proving that the situation is different from the one dimensional case.

Finally we compare our result to another representation theorem where the integration process is performed with respect to an operator valued measure.

**1.**J. Diestel and J. J. Uhl,*Vector measures*, Mathematical Surveys Number 15, AMS, 1977. MR**56:12216****2.**N. Dinculeanu,*Vector measures*, Pergamon Press, 1967. MR**34:6011b****3.**N. Dunford and J. Schwartz,*Linear operators*, 1988 printing. MR**90g:47001a**; MR**90g:47001b**; MR**90g:47001c****4.**E. Hille and R.S. Phillips,*Functional Analysis and Semi-groups*, Colloq. Publ. Amer. Math. Soc., 1957. MR**19:664d****5.**A. H. Shuchat,*Integral Representation Theorems in Topological Vector Spaces,*Trans. Amer. Math. Soc. 172 (1972), 373-397. MR**47:826**

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Additional Information

**Lakhdar Meziani**

Affiliation:
Department of Mathematics, Faculty of Science, University of Batna, Algeria

Email:
mezianilakhdar@hotmail.com

DOI:
https://doi.org/10.1090/S0002-9939-02-06336-0

Keywords:
Integral representation,
Riesz theorem,
Bochner integral

Received by editor(s):
October 5, 2000

Received by editor(s) in revised form:
February 10, 2001

Published electronically:
January 17, 2002

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2002
American Mathematical Society