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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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