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Characterizations of weakly compact operators on \(C_0(T)\). (English) Zbl 0906.47021

Summary: Let \(T\) be a locally compact Hausdorff space and let \(C_0(T)= \{f : T \rightarrow \mathbb{C}\), \(f\) is continuous and vanishes at infinity} be provided with the supremum norm. Let \(\mathcal{B}_c(T)\) and \(\mathcal{B}_0(T)\) 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 \(\mathcal{B}_0(T)\) are called the Baire sets of \(T\). \(M(T)\) denotes the dual of \(C_0(T)\). Let \(X\) be a quasicomplete locally convex Hausdorff space. Suppose \(u: C_0(T) \rightarrow X\) 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.

MSC:

47B38 Linear operators on function spaces (general)
46G10 Vector-valued measures and integration
28B05 Vector-valued set functions, measures and integrals
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