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On weakly compact operators on spaces of vector valued continuous functions

Author: Fernando Bombal
Journal: Proc. Amer. Math. Soc. 97 (1986), 93-96
MSC: Primary 47B05; Secondary 46B22, 46E40
MathSciNet review: 831394
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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.

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Article copyright: © Copyright 1986 American Mathematical Society

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