A stronger Dunford-Pettis property

H. Carrión; P. Galindo; M. L. Lourenço

Displaying similar documents to “A stronger Dunford-Pettis property”

On the class of order Dunford-Pettis operators

Khalid Bouras, Abdelmonaim El Kaddouri, Jawad H&#039;michane, Mohammed Moussa (2013)

Mathematica Bohemica

Similarity:

We characterize Banach lattices $E$ and $F$ on which the adjoint of each operator from $E$ into $F$ which is order Dunford-Pettis and weak Dunford-Pettis, is Dunford-Pettis. More precisely, we show that if $E$ and $F$ are two Banach lattices then each order Dunford-Pettis and weak Dunford-Pettis operator $T$ from $E$ into $F$ has an adjoint Dunford-Pettis operator $T^{'}$ from $F^{'}$ into $E^{'}$ if, and only if, the norm of $E^{'}$ is order continuous or $F^{'}$ has the Schur property. As a consequence we show that, if $E$ and $F$ are...

Completely Continuous operators

Ioana Ghenciu, Paul Lewis (2012)

Colloquium Mathematicae

Similarity:

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...

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

Abderrahman Retbi (2015)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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.

Copies of $ℓ_{\infty}$ in the space of Pettis integrable functions with integrals of finite variation

Juan Carlos Ferrando (2012)

Studia Mathematica

Similarity:

Let (Ω,Σ,μ) be a complete finite measure space and X a Banach space. We show that the space of all weakly μ-measurable (classes of scalarly equivalent) X-valued Pettis integrable functions with integrals of finite variation, equipped with the variation norm, contains a copy of $ℓ_{\infty}$ if and only if X does.

Correction to the paper ’Copies of $ℓ_{\infty}$ in the space of Pettis integrable functions with integrals of finite variation’ (Studia Math. 210 (2012), 93-98)

Juan Carlos Ferrando (2016)

Studia Mathematica

Similarity:

Order-bounded operators from vector-valued function spaces to Banach spaces

Marian Nowak (2005)

Banach Center Publications

Similarity:

Let E be an ideal of L⁰ over a σ-finite measure space (Ω,Σ,μ). For a real Banach space $(X, | | \cdot | |_{X})$ let E(X) be a subspace of the space L⁰(X) of μ-equivalence classes of strongly Σ-measurable functions f: Ω → X and consisting of all those f ∈ L⁰(X) for which the scalar function ${| | f (\cdot) | |}_{X}$ belongs to E. Let E(X)˜ stand for the order dual of E(X). For u ∈ E⁺ let $D_{u} (= {f \in E (X) : | | f (\cdot) | |}_{X} \leq u)$ stand for the order interval in E(X). For a real Banach space $(Y, | | \cdot | |_{Y})$ a linear operator T: E(X) → Y is said to be order-bounded whenever for each u ∈...

Extension and lifting of weakly continuous polynomials

Raffaella Cilia, Joaquín M. Gutiérrez (2005)

Studia Mathematica

Similarity:

We show that a Banach space X is an ℒ₁-space (respectively, an $ℒ_{\infty}$ -space) if and only if it has the lifting (respectively, the extension) property for polynomials which are weakly continuous on bounded sets. We also prove that X is an ℒ₁-space if and only if the space $_{w b} (^{m} X)$ of m-homogeneous scalar-valued polynomials on X which are weakly continuous on bounded sets is an $ℒ_{\infty}$ -space.

Dieudonné operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

Similarity:

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...

Dunford-Pettis operators on the space of Bochner integrable functions

Marian Nowak (2011)

Banach Center Publications

Similarity:

Let (Ω,Σ,μ) be a finite measure space and let X be a real Banach space. Let $L^{Φ} (X)$ be the Orlicz-Bochner space defined by a Young function Φ. We study the relationships between Dunford-Pettis operators T from L¹(X) to a Banach space Y and the compactness properties of the operators T restricted to $L^{Φ} (X)$ . In particular, it is shown that if X is a reflexive Banach space, then a bounded linear operator T:L¹(X) → Y is Dunford-Pettis if and only if T restricted to $L^{\infty} (X)$ is $(τ (L^{\infty} (X), L ¹ (X *)), | | \cdot | |_{Y})$ -compact.

On weak supercyclicity II

Carlos S. Kubrusly, Bhagwati P. Duggal (2018)

Czechoslovak Mathematical Journal

Similarity:

This paper considers weak supercyclicity for bounded linear operators on a normed space. On the one hand, weak supercyclicity is investigated for classes of Hilbert-space operators: (i) self-adjoint operators are not weakly supercyclic, (ii) diagonalizable operators are not weakly $l$ -sequentially supercyclic, and (iii) weak $l$ -sequential supercyclicity is preserved between a unitary operator and its adjoint. On the other hand, weak supercyclicity is investigated for classes of normed-space...

On the Dunford-Pettis property of tensor product spaces

Ioana Ghenciu (2011)

Colloquium Mathematicae

Similarity:

We give sufficient conditions on Banach spaces E and F so that their projective tensor product $E \otimes_{π} F$ and the duals of their projective and injective tensor products do not have the Dunford-Pettis property. We prove that if E* does not have the Schur property, F is infinite-dimensional, and every operator T:E* → F** is completely continuous, then $(E \otimes_{ϵ} F) *$ does not have the DPP. We also prove that if E* does not have the Schur property, F is infinite-dimensional, and every operator T: F** → E* is...

Some results on the class of $σ$ -unbounded Dunford-Pettis operators

Noufissa Hafidi, Jawad H'michane (2021)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We introduce and study the class of unbounded Dunford--Pettis operators. As consequences, we give basic properties and derive interesting results about the duality, domination problem and relationship with other known classes of operators.

Integration and decompositions of weak $^{*}$ -integrable multifunctions

Kazimierz Musiał (2025)

Czechoslovak Mathematical Journal

Similarity:

Conditions guaranteeing Pettis integrability of a Gelfand integrable multifunction and a decomposition theorem for the Henstock-Kurzweil-Gelfand integrable multifunctions are presented.

Application of $(L)$ sets to some classes of operators

Kamal El Fahri, Nabil Machrafi, Jawad H&#039;michane, Aziz Elbour (2016)

Mathematica Bohemica

Similarity:

The paper contains some applications of the notion of $Ł$ sets to several classes of operators on Banach lattices. In particular, we introduce and study the class of order $(L)$ -Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an $(L)$ sets. As a sequence characterization of such operators, we see that an operator $T : X \to E$ from a Banach space into a Banach lattice is order $Ł$ -Dunford-Pettis, if and only if $| T (x_{n}) | \to 0$ for $σ (E, E^{'})$ for every...

On projectional skeletons in Vašák spaces

Ondřej F. K. Kalenda (2017)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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.

Opérateurs sur un espace $L^{1}$

Alain Costé, Françoise Lust-Piquard (1987)

Colloquium Mathematicae

Similarity: