Dunford-Pettis Operators Research Articles

Some properties of weak* Dunford-Pettis operators on Banach lattices

We study relationships between the class of weak* Dunford-Pettis operators and other classes of operators like unbounded absolutely weakly Dunford-Pettis, weak Dunford-Pettis, and Dunford-Pettis operators.

Existence results for a nonlinear neutron transport equation with elastic and inelastic collision operators in L p -spaces

In this paper, we discuss the existence of solutions to a stationary neutron transport equation involving elastic and inelastic collision operators in L p - espaces ( 1 ≤ p < ∞ ) . For 1 < p < ∞ , we use the Krasnosel'slii fixed point theorem and the compactness which involved by the averaging result for neutron transport equation. For p = 1, our approach is different, it uses the measure of weak noncompactness of De Blasi, the concepts of Dunford–Pettis operators together with a recent version of Krasnosel'skii's fixed point theorem involving ws-compact and ww-compact operators and the weak compactness.

Some results on various types of compactness of weak* Dunford–Pettis operators on Banach lattices

AbstractWe study the relationship between weak* Dunford–Pettis and weakly (resp. M-weakly, order weakly, almost M-weakly, and almost L-weakly) operators on Banach lattices. The following is one of the major results dealing with this matter: If E and F are Banach lattices such that F is Dedekind $\sigma $ -complete, then each positive weak* Dunford–Pettis operator $T:E\rightarrow F$ is weakly compact if and only if one of the following assertions is valid: (a) the norms of $E^{\prime }$ and F are order continuous; (b) E is reflexive; and (c) F is reflexive.

Some results on the class of weakly sequentially precompact sets and operators

The paper contains some results on weakly sequentially precompact sets and operators. In particular, we establish some relationships between weakly sequentially precompat operators and those whose the adjoint map (L) sets into relatively norm compact ones. Besides, we characterize the class of weak* Dunford-Pettis operators through weakly sequentially precompact operators and deduce in the sequel a new characterization of Dunford-Pettis* property. Moreover, we generalize [9, Theorem 2.5.9] and show that order weakly compact operators carry almost order Dunford-Pettis sets into weakly sequentially precompact ones. Furthermore, we prove that the product of order weakly compact operators and b-weakly compact ones maps weakly sequentially precompact sets into relatively weakly compact ones. Finally, we present some results about the positive Schur property.

Large structures within the class of summing operators

We investigate lineability/spaceability problems within the setting of multilinear summing operators on quasi-Banach sequence spaces. Furthermore, we deal with the contemporary geometric notions of pointwise-lineability and (α,β)-lineability. Among other results, we prove that the class of multilinear operators taking values on ℓq(0<q≤∞) that are absolutely but not multiple summing is pointwise c-spaceable when non-empty. Spaceability in other classes, such as Dunford-Pettis operators and multilinear operators on general summable scalar families, is also studied.

The class of demi KB-operators on Banach lattices

In this paper, we introduce and study the new concept of demi KB-operators. Let $E$ be a Banach lattice. An operator $T: E\longrightarrow E$ is said to be a demi KB-operator if, for every positive increasing sequence $\{x_{n}\}$ in the closed unit ball $\mathcal{B}_{E}$ of $E$ such that $\{x_{n}-Tx_{n}\}$ is norm convergent to $x\in E$, there is a norm convergent subsequence of $\{x_{n}\}$. If the latter sequence has a weakly convergent subsequence then $T$ is called a weak demi KB-operator. We also investigate the relationship of these classes of operators with classical notions of operators, such as b-weakly demicompact operators and demi Dunford-Pettis operators.

(Non-)Dunford–Pettis operators on noncommutative symmetric spaces

We fully characterize the class of von Neumann algebras M (on a separable Hilbert space) such that any non-Dunford–Pettis operator on the predual M⁎ fixes a copy of ⊕ℓ1(ℓ2), which extends results due to Bourgain [9] and Rosenthal [49]. We also fully characterize those noncommutative symmetric spaces affiliated with different types of von Neumann algebras, which have the Dunford–Pettis property. This extends earlier results for the special cases (e.g., for symmetric function spaces, and von Neumann algebras and their preduals), by Bunce [11], by Chu and Iochum [15], and by Kaminska and Mastylo [36]. Finally, we supply new examples of symmetric spaces with the DPP on general (non-resonant) σ-finite measure spaces.

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

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.

On the class of demi Dunford–Pettis operators

On the class of demi Dunford–Pettis operators

Weak limited sets and operators on Banach lattices

In this paper, we prove that an operator T : E → F , between two Banach lattices, maps order intervals onto weak limited sets if and only if the modulus |ST| exists and is Dunford-Pettis for every Dunford-Pettis operator S : F → c0. Next, we establish that a Banach lattice E does not contain any isomorphic copy of ℓ 1 if and only if the order intervals of E are weak limited and the norm of E′ is order continuous. We also investigate the domination problem of the class of weak limited operators.

English

The present paper is devoted to some applications of the notion of L-Dunford-Pettis sets to several classes of operators on Banach lattices. More precisely, we establish some characterizations of weak Dunford-Pettis, Dunford-Pettis completely continuous, and weak almost Dunford-Pettis operators. Next, we study the relationships between LDunford-Pettis, and Dunford-Pettis (relatively compact) sets in topological dual Banach spaces.

Unbounded absolutely weak Dunford–Pettis operators

In the present article, we expose various properties of unbounded absolutely weak Dunford?Pettis and unbounded absolutely weak compact operators on a Banach lattice E. In addition to their topological and lattice properties, we investigate relationships between M-weakly compact operators, L-weakly compact operators, and order weakly compact operators with unbounded absolutely weak Dunford-Pettis operators. We show that the square of any positive uaw-Dunford-Pettis (M-weakly compact) operator on an order continuous Banach lattice is compact. Many examples are given to illustrate the essential conditions.

Some results on unbounded absolute weak Dunford–Pettis operators

In this paper, we characterize Banach lattices on which each Dunford-Pettis operator (or weak Dunford-Pettis) is unbounded absolute weak Dunford-Pettis operator and the converse.

A complement of positive weak almost Dunford-Pettis operators on Banach lattices

In this paper, we give some necessary and sufficient conditions such that each positive operator between two Banach lattices is weak almost Dunford-Pettis, and we derive some interesting results about the weak Dunford-Pettis property in Banach lattices.

The relationship between weak Dunford–Pettis operators and semi-compact operators

We investigate Banach lattices on which each positive weak Dunford–Pettis operator is semi-compact and conversely.

English

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 $\rm (L)$-Dunford-Pettis operators, that is, operators from a Banach space into a Banach lattice whose adjoint maps order bounded subsets to an $\rm (L)$ sets. As a sequence characterization of such operators, we see that an operator $T\colon X\rightarrow E$ from a Banach space into a Banach lattice is order $Ł$-Dunford-Pettis, if and only if $|T(x_{n})|\rightarrow 0$ for $\sigma (E,E')$ for every weakly null sequence $(x_{n})\subset X$. We also investigate relationships between order $Ł$-Dunford-Pettis, $\rm AM$-compact, weak* Dunford-Pettis, and Dunford-Pettis operators. In particular, it is established that each operator $T\colon E\rightarrow F$ between Banach lattices is Dunford-Pettis whenever it is both order $\rm (L)$-Dunford-Pettis and weak* Dunford-Pettis, if and only if $E$ has the Schur property or the norm of $F$ is order continuous.

Schur operators and domination problem

In this paper we are concerned with developing generalizing concepts of Dunford–Pettis operators analogous to the generalization of compact operators by strictly singular operators. Also, we give some new results concerning the domination problem in the setting of positive operators between Banach lattices.

Existence Results for a Nonlinear Transport Equation with Unbounded Admissible Velocities Space

In this paper, we discuss existence results for the nonlinear boundary value problem (1.1)–(1.2) on \({L^{1}}\)-spaces. This problem was already considered in Latrach and Zeghal (J Appl Math Comp 219:1163–1172, 2012) under the hypothesis that the velocity space has finite measure. This condition was used to establish a compactness result necessary to use fixed point theorems. In this work we show that this hypothesis is not necessary and it can be relaxed. Our analysis uses a new measure of weak noncompactness adapted to the problem, the concept of Dunford–Pettis operators and a new version of Darbo’s fixed point theorem.

Riemann integrability versus weak continuity

In this paper we focus on the relation between Riemann integrability and weak continuity. A Banach space X is said to have the weak Lebesgue property if every Riemann integrable function from [0,1] into X is weakly continuous almost everywhere. We prove that the weak Lebesgue property is stable under ℓ1-sums and obtain new examples of Banach spaces with and without this property. Furthermore, we characterize Dunford–Pettis operators in terms of Riemann integrability and provide a quantitative result about the size of the set of τ-continuous nonRiemann integrable functions, with τ a locally convex topology weaker than the norm topology.

Optimal extensions of compactness properties for operators on Banach function spaces

Compactness type properties for operators acting in Banach function spaces are not always preserved when the operator is extended to a bigger space. Moreover, it is known that there exists a maximal (weakly) compact linear extension of a (weakly) compact operator if and only if its maximal continuous linear extension to its optimal domain is (weakly) compact. We show that the same happens if we consider AM-compactness for the operator, and we give some partial results regarding Dunford–Pettis operators. Narrow operators—considered as a family defined by a weak compactness type property—are also analyzed from this point of view. Finally, we provide some applications of the fact that an operator from a Banach function space extends to a narrow operator if and only if it is narrow.