Semi-compactness of positive Dunford–Pettis operators on Banach lattices

Abstract. We study the semi-compactness of positive Almost Dunford-Pettis (resp. the square of positive Almost Dunford-Pettis) operators on Banach lattices and we give some consequences. Introduction and notationRecall from [7] that an operator T from a Banach lattice E into a Banach space F is said to be almost Dunford-Pettis if the sequence ( T (x n ) ) converges to 0 for every weakly null sequence (x n ) consisting of pairwise disjoint elements in E. An operator between two Banach spaces is called Dunford-Pettis, if it maps weakly null sequences into norm null sequences. It is evident that every Dunford-Pettis operator from a Banach lattice E into a Banach space F is almost Dunford-Pettis, but the converse is false in general. In fact, the identity operatoris almost Dunford-Pettis, but it is not Dunford-Pettis. In this paper, we introduce a new notion that we call σ-almost order bounded subsets of a Banach lattice. It is a weaker notion than the usual notion of almost order bounded subset, introduced by Zaanen in [9]. And we use this notion to study the semi-compactness of almost Dunford-Pettis operators by characterizing Banach lattices on which each positive almost Dunford-Pettis operator is semicompact. In fact, we will prove that if E and F are two Banach lattices such that each positive almost Dunford-Pettis (resp. Dunford-Pettis) operator T : E → F is semi-compact, then the norm of E ′ is order continuous or the closed unit ball B F of F is σ-almost order bounded (and hence F ′ has the positive Schur property). As interesting consequences, we will give several characterizations of the order continuity of the dual norm. Finally,

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“…Also, this result generalizes the implication (3) =⇒ (1) of Theorem 2.2 of [2]. In fact, we have Theorem 2.5.…”

Section: Example It Follows From Proposition 24 That the Closed Unisupporting

confidence: 80%

Semi-compactness of almost Dunford–Pettis operators on Banach lattices

Aqzzouz

Elbour

2011

Demonstratio Mathematica

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“…As we asked in [6], a semi-compact operator is not necessary Dunford-Pettis, and conversely a Dunford-Pettis operator is not necessary semi-compact. For example, the identity operator Id l 1 : l 1 → l 1 is Dunford-Pettis but it is not semi-compact and conversely, the identity operator Id c : c → c is semi-compact but it is not Dunford-Pettis where c is the Banach lattice of all convergent sequences.…”

Section: Corollary 23 Let E and F Be Two Banach Lattices If The Nomentioning

confidence: 92%

Some properties of the class of positive Dunford–Pettis operators

Aqzzouz

Elbour

H’michane

2009

Journal of Mathematical Analysis and Applications

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“…And in the paper [4], Banach lattices on which each positive Dunford-Pettis operator is semi-compact, were investigated. Conversely, a semi-compact operator is not necessary Dunford-Pettis.…”

Section: The Dunford-pettis Property Of Semi-compact Operatorsmentioning

confidence: 99%

“…For the converse of results of [4], we will show that if E and F are two Banach lattices such that F is Dedekind σ-complete (resp. the norm of E is order continuous) and F is discrete or E has the Dunford-Pettis property, then each semi-compact operator T : E → F is Dunford-Pettis if and only if E has the Schur property or the norm of F is order continuous.…”

Section: Introduction and Notationmentioning

confidence: 96%

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The almost Dunford–Pettis property of semi-compact operators

Aqzzouz

Elbour

2012

Demonstratio Mathematica

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Abstract. We establish necessary and sufficient conditions for which each positive semi-compact operator (resp. the second power of a positive semi-compact operator) is almost Dunford-Pettis (resp. Dunford-Pettis). Introduction and notationRecall from [12] that an operator T from a Banach lattice E into a Banach space F is said to be almost Dunford-Pettis if the sequence ( T (x n ) ) converges to 0 for every weakly null sequence (x n ) consisting of pairwise disjoint elements in E. An operator T from a Banach space E into another F is said to be Dunford-Pettis if it carries weakly compact subsets of E onto compact subsets of F . It is clear that every Dunford-Pettis operator from a Banach lattice into a Banach space is almost Dunford-Pettis but the converse is false in general. In fact, the identity operatoris almost Dunford-Pettis, but it is not Dunford-Pettis.In [5] (resp. [4]), the semi-compactness of positive almost Dunford-Pettis (resp. Dunford-Pettis) operators on Banach lattices is studied, but the reciprocal, i.e. the almost Dunford-Pettis (resp. Dunford-Pettis) property of semi-compact operators, is not yet done. The goal of this paper is to make this study, by characterizing Banach lattices for which each semi-compact operator is almost Dunford-Pettis. As consequences, we obtain some interesting results on the Dunford-Pettis property of semi-compact operators.More precisely, we will prove that if E and F are two Banach lattices such that F is Dedekind σ-complete (resp. the norm of E is order continuous), then each regular semi-compact operator T : E → F is almost DunfordPettis if and only if E has the positive Schur property or the norm of F is 2000 Mathematics Subject Classification: 46A40, 46B40, 46B42.

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Semi-compactness of positive Dunford–Pettis operators on Banach lattices

Abstract: Abstract. We investigate Banach lattices on which each positive DunfordPettis operator is semi-compact and the converse. As an interesting consequence, we obtain Theorem 2.7 of Aliprantis-Burkinshaw and an element of Theorem 1 of Wickstead.

Cited by 8 publications

References 11 publications

Semi-compactness of almost Dunford–Pettis operators on Banach lattices

Semi-compactness of almost Dunford–Pettis operators on Banach lattices

Some properties of the class of positive Dunford–Pettis operators

The almost Dunford–Pettis property of semi-compact operators

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