On weakly compact operators on spaces of vector valued continuous functions

Bombal, Fernando

doi:10.1090/S0002-9939-1986-0831394-3

On weakly compact operators on spaces of vector valued continuous functions
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Proc. Amer. Math. Soc. 97 (1986), 93-96 Request permission

Abstract:

Let $K$ and $S$ be compact Hausdorff spaces and $\theta$ a continuous function from $K$ onto $S$. Then for any Banach space $E$ the map $f \mapsto f \circ \theta$ isometrically embeds $C(S,E)$ as a closed subspace of $C(K,E)$. In this note we prove that when $E’$ has the Radon-Nikodým property, every weakly compact operator on $C(S,E)$ can be lifted to a weakly compact operator on $C(K,E)$. As a consequence, we prove that the compact dispersed spaces $K$ are characterized by the fact that $C(K,E)$ has the Dunford-Pettis property whenever $E$ has.

References

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Some classes of operators on

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