Characterizations of weakly compact operators on $C_o(T)$

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 | References | Similar Articles | Additional Information

Abstract: Let $T$ be a locally compact Hausdorff space and let $C_o(T)= \{f : T \rightarrow \mathbb {C}$, $f$ is continuous and vanishes at infinity} be provided with the supremum norm. Let $\mathcal {B}_c(T)$ and $\mathcal {B}_o(T)$ be the $\sigma$-rings generated by the compact subsets and by the compact $G_\delta$ subsets of $T$, respectively. The members of $\mathcal {B}_c(T)$ are called $\sigma$-Borel sets of $T$ since they are precisely the $\sigma$-bounded Borel sets of $T$. The members of $\mathcal {B}_o(T)$ are called the Baire sets of $T$. $M(T)$ denotes the dual of $C_o(T)$. Let $X$ be a quasicomplete locally convex Hausdorff space. Suppose $u: C_o(T) \rightarrow X$ is a continuous linear operator. Using the Baire and $\sigma$-Borel characterizations of weakly compact sets in $M(T)$ 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 $u$ to be weakly compact, several of which are new. The independent results on the regularity and on the regular Borel extendability of $\sigma$-additive $X$-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

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