On the class of order Dunford-Pettis operators

Khalid Bouras; Abdelmonaim El Kaddouri; Jawad H'michane; Mohammed Moussa

Displaying similar documents to “On the class of order Dunford-Pettis operators”

Completely Continuous operators

Ioana Ghenciu, Paul Lewis (2012)

Colloquium Mathematicae

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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...

A stronger Dunford-Pettis property

H. Carrión, P. Galindo, M. L. Lourenço (2008)

Studia Mathematica

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We discuss a strong version of the Dunford-Pettis property, earlier named (DP*) property, which is shared by both ℓ₁ and $ℓ_{\infty}$ . It is equivalent to the Dunford-Pettis property plus the fact that every quotient map onto c₀ is completely continuous. Other weak sequential continuity results on polynomials and analytic mappings related to the (DP*) property are shown.

Weakly precompact subsets of L₁(μ,X)

Ioana Ghenciu (2012)

Colloquium Mathematicae

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Let (Ω,Σ,μ) be a probability space, X a Banach space, and L₁(μ,X) the Banach space of Bochner integrable functions f:Ω → X. Let W = f ∈ L₁(μ,X): for a.e. ω ∈ Ω, ||f(ω)|| ≤ 1. In this paper we characterize the weakly precompact subsets of L₁(μ,X). We prove that a bounded subset A of L₁(μ,X) is weakly precompact if and only if A is uniformly integrable and for any sequence (fₙ) in A, there exists a sequence (gₙ) with $g ₙ \in c o f_{i} : i \geq n$ for each n such that for a.e. ω ∈ Ω, the sequence (gₙ(ω)) is weakly...

On projectional skeletons in Vašák spaces

Ondřej F. K. Kalenda (2017)

Commentationes Mathematicae Universitatis Carolinae

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We provide an alternative proof of the theorem saying that any Vašák (or, weakly countably determined) Banach space admits a full $1$ -projectional skeleton. The proof is done with the use of the method of elementary submodels and is comparably simple as the proof given by W. Kubiś (2009) in case of weakly compactly generated spaces.

On Pettis integrability

Juan Carlos Ferrando (2003)

Czechoslovak Mathematical Journal

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Assuming that $(Ω, Σ, μ)$ is a complete probability space and $X$ a Banach space, in this paper we investigate the problem of the $X$ -inheritance of certain copies of $c_{0}$ or $ℓ_{\infty}$ in the linear space of all [classes of] $X$ -valued $μ$ -weakly measurable Pettis integrable functions equipped with the usual semivariation norm.

On some subsets of $L_{1} (μ, E)$

Fernando Bombal (1991)

Czechoslovak Mathematical Journal

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On the class of positive almost weak $^{☆}$ Dunford-Pettis operators

Abderrahman Retbi (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we introduce and study the class of almost weak $^{☆}$ Dunford-Pettis operators. As consequences, we derive the following interesting results: the domination property of this class of operators and characterizations of the wDP $^{☆}$ property. Next, we characterize pairs of Banach lattices for which each positive almost weak $^{☆}$ Dunford-Pettis operator is almost Dunford-Pettis.

Dieudonné operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

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A bounded linear operator between Banach spaces is called a Dieudonné operator ( = weakly completely continuous operator) if it maps weakly Cauchy sequences to weakly convergent sequences. Let (Ω,Σ,μ) be a finite measure space, and let X and Y be Banach spaces. We study Dieudonné operators T: L¹(X) → Y. Let $i_{\infty} : L^{\infty} (X) \to L ¹ (X)$ stand for the canonical injection. We show that if X is almost reflexive and T: L¹(X) → Y is a Dieudonné operator, then $T \circ i_{\infty} : L^{\infty} (X) \to Y$ is a weakly compact operator. Moreover, we obtain that...

The weak compactness of almost Dunford-Pettis operators

Belmesnaoui Aqzzouz, Aziz Elbour, Othman Aboutafail (2011)

Commentationes Mathematicae Universitatis Carolinae

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We characterize Banach lattices on which every positive almost Dunford-Pettis operator is weakly compact.

Weak vs. norm compactness in $L_{1}$ : the Bocce criterion

Maria Girardi (1991)

Studia Mathematica

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