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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
MathSciNet review: 1615942
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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

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