Representation of set valued operators

Papageorgiou, Nikolaos S.

doi:10.1090/S0002-9947-1985-0808737-3

Representation of set valued operators

Author: Nikolaos S. Papageorgiou
Journal: Trans. Amer. Math. Soc. 292 (1985), 557-572
MSC: Primary 47H99; Secondary 28B20, 46E30, 46G99
DOI: https://doi.org/10.1090/S0002-9947-1985-0808737-3
MathSciNet review: 808737
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Abstract: In this paper we prove representation theorems for set valued additive operators acting on the spaces $L_X^1(X = {\text {separable Banach space)}}$, ${L^1}$ and ${L^\infty }$. Those results generalize well-known ones for single valued operators and among them the celebrated Dunford-Pettis theorem. The properties of these representing integrals are studied. We also have a differentiability result for multifunctions analogous to the one that says that an absolutely continuous function from a closed interval into a Banach space with the Radon-Nikodým property is almost everywhere differentiable and also it is the primitive of its strong derivative. Finally we have a necessary and sufficient condition for the set of integrable selectors of a multifunction to be $w$-compact in $L_X^1$. This result is a new very general result about $w$-compactness in the Lebesgue-Bochner space $L_X^1$.

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