Characterizations of

weakly compact operators on

Author:
T. V. Panchapagesan

Journal:
Trans. Amer. Math. Soc. **350** (1998), 4849-4867

MSC (1991):
Primary 47B38, 46G10; Secondary 28B05

DOI:
https://doi.org/10.1090/S0002-9947-98-02358-7

MathSciNet review:
1615942

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Abstract: Let be a locally compact Hausdorff space and let , is continuous and vanishes at infinity} be provided with the supremum norm. Let and be the -rings generated by the compact subsets and by the compact subsets of , respectively. The members of are called -Borel sets of since they are precisely the -bounded Borel sets of . The members of are called the Baire sets of . denotes the dual of . Let be a quasicomplete locally convex Hausdorff space. Suppose is a continuous linear operator. Using the Baire and -Borel characterizations of weakly compact sets in as given in a previous paper of the author's and combining the integration technique of Bartle, Dunford and Schwartz, we obtain 35 characterizations for the operator to be weakly compact, several of which are new. The independent results on the regularity and on the regular Borel extendability of -additive -valued Baire measures are deduced as an immediate consequence of these characterizations. Some other applications are also included.

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Additional Information

**T. V. Panchapagesan**

Affiliation:
Departamento de Matemáticas, Facultad de Ciencias, Universidad de Los Andes, Mérida, Venezuela

Email:
panchapa@ciens.ula.ve

DOI:
https://doi.org/10.1090/S0002-9947-98-02358-7

Keywords:
Weakly compact operators,
representing measure,
vector measure,
quasicomplete locally compact Hausdorff space,
Borel (resp. $\sigma$-Borel,
Baire) regularity,
inner regularity and outer regularity

Received by editor(s):
November 17, 1995

Additional Notes:
Supported by the C.D.C.H.T. project C-586 of the Universidad de los Andes, Mérida, and by the international cooperation project between CONICIT-Venezuela and CNR-Italy.

Dedicated:
Dedicated to Professor V. K. Balachandran on the occasion of his seventieth birthday

Article copyright:
© Copyright 1998
American Mathematical Society