Representation of Set Valued Operators
Copyright policy )Representation of Set Valued Operators
In this paper we prove representation theorems for set valued additive operators acting on the spaces L X 1 ( X = separable Banach space) L_X^1(X = {\text {separable Banach space)}} , L 1 {L^1} and L ∞ {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 w -compact in L X 1 L_X^1 . This result is a new very general result about w w -compactness in the Lebesgue-Bochner space L X 1 L_X^1 .
Dunford-Pettis theorem, Lebesgue-Bochner space, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), weak compactness, Spaces of vector- and operator-valued functions, representation theorem for set valued additive operators, absolutely continuous set valued operators, Linear operators on function spaces (general), Vector-valued set functions, measures and integrals, Vector-valued measures and integration, Set-valued set functions and measures; integration of set-valued functions; measurable selections, Radon-Nikodým, Kreĭn-Milman and related properties, Compactness in topological linear spaces; angelic spaces, etc., multivalued operators, integrable selectors of an integrably bounded multifunction
Dunford-Pettis theorem, Lebesgue-Bochner space, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), weak compactness, Spaces of vector- and operator-valued functions, representation theorem for set valued additive operators, absolutely continuous set valued operators, Linear operators on function spaces (general), Vector-valued set functions, measures and integrals, Vector-valued measures and integration, Set-valued set functions and measures; integration of set-valued functions; measurable selections, Radon-Nikodým, Kreĭn-Milman and related properties, Compactness in topological linear spaces; angelic spaces, etc., multivalued operators, integrable selectors of an integrably bounded multifunction
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