A note on the positive Schur property

Wnuk, Witold

doi:10.1017/s0017089500007692

Cited by 27 publications

(22 citation statements)

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“…Tite erder continuity of the norm imphies tite follewing fact (see [15] Lemma 1): even' Dunferd-Pettis operator T-E--*c, is a difference of twe pesitive eperaters. TI-¡erefere even' continuous linear operater from E into 4 is regular, and so E itas te be discrete ( [14]).…”

Section: ¡9mentioning

confidence: 95%

“…Besides tite Seitur property ene can censider a weaker preperty called tite pesitive Scitur property ( [15]), i.e., 0-cx -*0 weakly implies x,,-*0 in nerm. A simple example ofa Banacit ¡attice witit tite positive Scitur property and wititout tite Scitur preperty is a space L'(ji), witere pi is net purely atemic.…”

Section: Remárks On Banách Lattices Witii the Positive Schur Propertymentioning

confidence: 99%

“…Tite last pan of titis paper centains a few remarks concerning Banacb ¡at-tices having tite so-called pesitive Scitur property (i.e., Oc x,,-*0 weakly implies x 0-*0 in nenn, see [15]). …”

Section: Introductionmentioning

confidence: 99%

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Some characterizations of Banach lattices with the Schur property

Wnuk¹

1989

Rev. Mat. Complut.

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ABSTRACT. This note centalas a short proef ofIhe equivalence of ihe Sehur and Dunford-Pettis properties in ihe class of discrete KB-spaces. Wc also present an aperator characterization of dic Schur preperty (Theorezn 2) and we natice thai Banach laitices which are band hercditary 1' coincide wiah Banach lattices having tIte Schur property (this characterizatian is due te 1. Papa [lO]).Moreover, the paper affers examples of Banach laitices with dic positive Schur property and without tite Schur preperty and which are noa isamarphic to any AL-spacc

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Section: ¡9mentioning

confidence: 95%

Section: Remárks On Banách Lattices Witii the Positive Schur Propertymentioning

confidence: 99%

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Some characterizations of Banach lattices with the Schur property

Wnuk¹

1989

Rev. Mat. Complut.

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“…For example, the Banach lattice L 1 ([0, 1]) has the positive Schur property but does not have the Schur property. For more information about this notion see [6].…”

Section: And Hence T (Bmentioning

confidence: 99%

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

Aqzzouz

Elbour

2011

Demonstratio Mathematica

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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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“…In fact, it follows from Josefson-Nissenzweig theorem [9,11] that the existence of a sequence (x n ) of the topological dual of c such that x n = 1 for all n and x n −→ 0 for the weak topology σ (c , c). We consider the operator T, which we can find in Wnuk ( [15], p. 170), defined by…”

Section: Lemma 24 There Exists An Operator From C Into C 0 That Is mentioning

confidence: 99%

The Duality Problem for the Class of Order Weakly Compact Operators

Aqzzouz¹,

H’michane

2009

Glasgow Math. J.

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Abstract. We study the duality problem for order weakly compact operators by giving sufficient and necessary conditions under which the order weak compactness of an operator implies the order weak compactness of its adjoint and conversely.2000 AMS Classification. 46A40; 46B40; 46B42. considered order weakly compact operators from a vector-valued function space into a Banach space and gave a characterization of an order weakly compact operator T in terms of the continuity of its adjoint relatively to some weak topologies. Introduction and notation.Let us recall that an operator T from a Banach lattice E into a Banach space F is said to be order weakly compact if for each x ∈ E + , the subset T ([0, x]) is relatively weakly compact in F, where E + = {x ∈ E : 0 ≤ x}.Contrarily to weakly compact operators [2, 13], the class of order weakly compact operators satisfies the domination problem. Indeed, if S and T are two operators from a Banach lattice E into another F such that 0 ≤ S ≤ T and T is order weakly compact, then S is order weakly compact [3].Also, the class of order weakly compact operators does not satisfy the duality property, that is, there exist order weakly compact operators whose adjoints are not order weakly compact. In fact, the identity operator of the Banach lattice l 1 is order weakly compact, but its adjoint, which is the identity operator of the Banach lattice l ∞ , is not order weakly compact. And conversely, there exist operators that are not order weakly compact but their adjoints are order weakly compact. In fact, the identity operator of the Banach lattice l ∞ is not order weakly compact but its adjoint, which is the identity operator of the topological dual (l ∞ ) , is order weakly compact.In [16], Zaanen investigated the duality problem for semi-compact operators. Also, in [5] and [16], the duality problem of AM-compact operators on Banach lattices was studied. They gave sufficient and necessary conditions for which the AM-compactness

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A note on the positive Schur property

Cited by 27 publications

References 4 publications

Some characterizations of Banach lattices with the Schur property

Some characterizations of Banach lattices with the Schur property

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

The Duality Problem for the Class of Order Weakly Compact Operators

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