Dunford-Pettis operators and weak Radon-Nikodým sets

Riddle, Lawrence H.

doi:10.1090/S0002-9939-1984-0740180-2

Dunford-Pettis operators and weak Radon-Nikodým sets
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by Lawrence H. Riddle

Proc. Amer. Math. Soc. 91 (1984), 254-256

DOI: https://doi.org/10.1090/S0002-9939-1984-0740180-2

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Abstract:

Let $K$ be a weak*-compact convex subset of a Banach space $X$. If every Dunford-Pettis operator from ${L_1}\left [ {0,1} \right ]$ into ${X^ * }$ that maps the set $\{ \chi E/\mu (E):E \text {measurable}, \mu (E) > 0\}$ into $K$ has a Pettis derivative, then $K$ is a weak Radon-Nikodým set. This positive answer to a question of M. Talagrand localizes a result of E. Saab.

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