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

Noufissa Hafidi; Jawad H'michane

Displaying similar documents to “Some results on the class of $σ$ -unbounded Dunford-Pettis operators”

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...

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.

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...

Absolutely continuous linear operators on Köthe-Bochner spaces

(2011)

Banach Center Publications

Similarity:

Let E be a Banach function space over a finite and atomless measure space (Ω,Σ,μ) and let $(X, | | \cdot | |_{X})$ and $(Y, | | \cdot | |_{Y})$ be real Banach spaces. A linear operator T acting from the Köthe-Bochner space E(X) to Y is said to be absolutely continuous if $| | T (1_{A ₙ} f) {| |}_{Y} \to 0$ whenever μ(Aₙ) → 0, (Aₙ) ⊂ Σ. In this paper we examine absolutely continuous operators from E(X) to Y. Moreover, we establish relationships between different classes of linear operators from E(X) to Y.

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...

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 ∈...

Opérateurs sur un espace $L^{1}$

Alain Costé, Françoise Lust-Piquard (1987)

Colloquium Mathematicae

Similarity:

On complemented copies of $c_{0}$ in spaces of operators, II

Giovanni Emmanuele (1994)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We show that as soon as $c_{0}$ embeds complementably into the space of all weakly compact operators from $X$ to $Y$ , then it must live either in $X^{*}$ or in $Y$ .

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 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.

Representing non-weakly compact operators

Manuel González, Eero Saksman, Hans-Olav Tylli (1995)

Studia Mathematica

Similarity:

For each S ∈ L(E) (with E a Banach space) the operator R(S) ∈ L(E**/E) is defined by R(S)(x** + E) = S**x** + E(x** ∈ E**). We study mapping properties of the correspondence S → R(S), which provides a representation R of the weak Calkin algebra L(E)/W(E) (here W(E) denotes the weakly compact operators on E). Our results display strongly varying behaviour of R. For instance, there are no non-zero compact operators in Im(R) in the case of $L^{1}$ and C(0,1), but R(L(E)/W(E)) identifies isometrically...

The Maurey extension property for Banach spaces with the Gordon-Lewis property and related structures

P. G. Casazza, N. J. Nielsen (2003)

Studia Mathematica

Similarity:

The main result of this paper states that if a Banach space X has the property that every bounded operator from an arbitrary subspace of X into an arbitrary Banach space of cotype 2 extends to a bounded operator on X, then every operator from X to an L₁-space factors through a Hilbert space, or equivalently $B (ℓ_{\infty}, X *) = Π ₂ (ℓ_{\infty}, X *)$ . If in addition X has the Gaussian average property, then it is of type 2. This implies that the same conclusion holds if X has the Gordon-Lewis property (in particular X could...

Asymptotic behaviour of the iterates of nonnegative operators on a Banach lattice

Jolanta Socała (1998)

Annales Polonici Mathematici

Similarity:

Asymptotic convergence theorems for nonnegative operators on Banach lattices, on $L^{\infty}$ , on C(X) and on $L^{p} (1 \leq p < \infty)$ are proved. The general results are applied to a class of integral operators on L¹.

Displaying similar documents to “Some results on the class of σ -unbounded Dunford-Pettis operators”

Displaying similar documents to “Some results on the class of $σ$ -unbounded Dunford-Pettis operators”