Let $$\,X\,$$ X be a completely regular Hausdorff space and $$\,\mathcal {B}o\,$$ B o be the $$\sigma $$ σ -algebra of Borel sets in X. Let $$C_b(X)$$ C b ( X ) (resp. $$B(\mathcal {B}o)$$ B ( B o ) ) be the space of all bounded continuous (resp. bounded $$\mathcal {B}o$$ B o -measurable) scalar functions on X, equipped with the natural strict topology $$\beta $$ β . We develop a general integral representation theory of $$(\beta ,\xi )$$ ( β , ξ ) -continuous operators from $$C_b(X)$$ C b ( X ) to a lcHs $$(E,\xi )$$ ( E , ξ ) with respect to the representing Borel measure taking values in the bidual $$E''_\xi $$ E ξ ′ ′ of $$(E,\xi )$$ ( E , ξ ) . It is shown that every $$(\beta ,\xi )$$ ( β , ξ ) -continuous operator $$T:C_b(X)\rightarrow E$$ T : C b ( X ) → E possesses a $$(\beta ,\xi _\mathcal {E})$$ ( β , ξ E ) -continuous extension $${\hat{T}}:B(\mathcal {B}o)\rightarrow E''_\xi $$ T ^ : B ( B o ) → E ξ ′ ′ , where $$\xi _\mathcal {E}$$ ξ E stands for the natural topology on $$E''_\xi $$ E ξ ′ ′ . If, in particular, X is a k-space and $$(E,\xi )$$ ( E , ξ ) is quasicomplete, we present equivalent conditions for a $$(\beta ,\xi )$$ ( β , ξ ) -continuous operator $$T:C_b(X)\rightarrow E$$ T : C b ( X ) → E to be weakly compact. As an application, we have shown that if X is a k-space and a quasicomplete lcHs $$(E,\xi )$$ ( E , ξ ) contains no isomorphic copy of $$c_0$$ c 0 , then every $$(\beta ,\xi )$$ ( β , ξ ) -continuous operator $$T:C_b(X)\rightarrow E$$ T : C b ( X ) → E is weakly compact.
Results in Mathematics – Springer Journals
Published: Apr 1, 2017
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