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Representation of set valued operators. (English) Zbl 0605.46037
This paper studies multivalued operators defined on a separable Banach space. It starts with a representation theorem for set valued additive operators acting on the Lebesgue-Bochner space $$L^ 1(X)$$. Then it has two more representation theorems for set valued operators acting on $$L^ 1$$ and $$L^{\infty}$$. Those results extend well known single valued ones, among them the celebrated Dunford-Pettis theorem. Then there is a differentiability result for absolutely continuous set valued operators and finally a weak compactness result for the set of integrable selectors of an integrably bounded multifunction.

##### MSC:
 46G10 Vector-valued measures and integration 47B38 Linear operators on function spaces (general) 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 46A50 Compactness in topological linear spaces; angelic spaces, etc. 46B22 Radon-Nikodým, Kreĭn-Milman and related properties 28B05 Vector-valued set functions, measures and integrals 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E40 Spaces of vector- and operator-valued functions
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##### References:
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