## 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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### References:

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