Variational semi-regularity and norm convergence.
Mapping Properties of
Bessaga and Pełczyński showed that if embeds in the dual of a Banach space X, then embeds as a complemented subspace of X. Pełczyński proved that every infinite-dimensional closed linear subspace of contains a copy of that is complemented in . Later, Kadec and Pełczyński proved that every non-reflexive closed linear subspace of contains a copy of that is complemented in . In this note a traditional sliding hump argument is used to establish a simple mapping property of which simultaneously...
The Dunford-Pettis property, the Gelfand-Phillips property, and L-sets
The Dunford-Pettis property and the Gelfand-Phillips property are studied in the context of spaces of operators. The idea of L-sets is used to give a dual characterization of the Dunford-Pettis property.
Completely Continuous operators
A Banach space X has the Dunford-Pettis property (DPP) provided that every weakly compact operator T from X to any Banach space Y is completely continuous (or a Dunford-Pettis operator). It is known that X has the DPP if and only if every weakly null sequence in X is a Dunford-Pettis subset of X. In this paper we give equivalent characterizations of Banach spaces X such that every weakly Cauchy sequence in X is a limited subset of X. We prove that every operator T: X → c₀ is completely continuous...
The Embeddability of c₀ in Spaces of Operators
Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space , as well as the complementation of the space of w*-w compact operators in the space of w*-w operators from X* to Y.
Almost Weakly Compact Operators
Dunford-Pettis type properties are studied in individual Banach spaces as well as in spaces of operators. Bibasic sequences are used to characterize Banach spaces which fail to have the Dunford-Pettis property. The question of whether a space of operators has a Dunford-Pettis property when the dual of the domain and the codomain have the respective property is studied. The notion of an almost weakly compact operator plays a consistent and important role in this study.
Evaluation maps, restriction maps, and compactness
Linear Operators and Vector Measures. II.
Vector Measures, c₀, and (sb) Operators
Emmanuele showed that if Σ is a σ-algebra of sets, X is a Banach space, and μ: Σ → X is countably additive with finite variation, then μ(Σ) is a Dunford-Pettis set. An extension of this theorem to the setting of bounded and finitely additive vector measures is established. A new characterization of strongly bounded operators on abstract continuous function spaces is given. This characterization motivates the study of the set of (sb) operators. This class of maps is used to extend results of P. Saab...
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