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Operators and the space of integrable scalar functions with respect to a Fréchet-valued measure

Part of: Topological linear spaces and related structures Special classes of linear operators Measures, integration, derivative, holomorphy

Published online by Cambridge University Press:  09 April 2009

Antonio Fernández  and
Francisco Naranjo
Antonio Fernández
Affiliation:
Department Matemática Aplicada II, Escuela Superior de Ingenieros, Camino de los Descubrimientos, s/n 41092-Sevilla, Spain e-mail: anfercar@matinc.us.es
Francisco Naranjo
Affiliation:
Department Matemática Aplicada II, Escuela Superior de Ingenieros, Camino de los Descubrimientos, s/n 41092-Sevilla, Spain e-mail: anfercar@matinc.us.es
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Abstract

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We consider the space L1 (ν, X) of all real functions that are integrable with respect to a measure v with values in a real Fréchet space X. We study L-weak compactness in this space. We consider the problem of the relationship between the existence of copies of l∞ in the space of all linear continuous operators from a complete DF-space Y to a Fréchet lattice E with the Lebesgue property and the coincidence of this space with some ideal of compact operators. We give sufficient conditions on the measure ν and the space X that imply that L1 (ν, X) has the Dunford-Pettis property. Applications of these results to Fréchet AL-spaces and Köthe sequence spaces are also given.

Keywords

Fréchet latticeL-weak compactnessDunford-Pettis propertyL-weakly compact operatorsgeneralized AL-spacesKöthe sequences spaces

MSC classification

Secondary: 46A04: Locally convex Fréchet spaces and (DF)-spaces 46A40: Ordered topological linear spaces, vector lattices 46G10: Vector-valued measures and integration 47B07: Operators defined by compactness properties
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 2 , October 1998 , pp. 176 - 193
Copyright
Copyright © Australian Mathematical Society 1998

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