Converses for the Dodds–Fremlin and Kalton–Saab theorems

Wickstead, Anthony

doi:10.1017/s0305004100074752

Cited by 83 publications

(77 citation statements)

References 7 publications

(11 reference statements)

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Order By: Relevance

“…If the norm of E is not order continuous, then it follows from the proof of Theorem 1 of Wickstead [9], that E contains a sublattice that is isomorphic to l 1 and there exists a positive projection P from E into l 1 . Consider the operator product…”

Section: Each Positive Dunford-pettis Operator From E Into F Is Semi-mentioning

confidence: 99%

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

Aqzzouz¹,

Nouira

Zraoula

2008

Proc. Amer. Math. Soc.

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Section: Each Positive Dunford-pettis Operator From E Into F Is Semi-mentioning

confidence: 99%

“…DunfordPettis) operators does not satisfy the analogue of Schauder's Theorem. However, the class of semi-compact operators satisfies the domination problem (Theorem 18.20 of [3]), but the class of Dunford-Pettis operators fails to satisfy this property, as was proved in [1], [7] and [9].…”

Section: Introduction and Notationmentioning

confidence: 96%

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

Aqzzouz¹,

Nouira

Zraoula

2008

Proc. Amer. Math. Soc.

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“…Supposons que la norme du treillis de Banach E n'est pas continue pour l'ordre. Alors il résulte de la démonstration du Théorème 1 de [10] que E contient un sous-treillis vectoriel isomorphe (en ordre) au treillis de Banach l 1 , et il existe aussi une projection positive P : E −→ l 1 . Il est clair que l'opérateur P est de Dunford-Pettis, mais l'opérateur P 2 = P n'est pas compact.…”

Section: Le Resultat Principalunclassified

Sur les opérateurs de Dunford-Pettis positifs qui sont faiblement compacts

Aqzzouz

Nouira

Zraoula

2005

Proc. Amer. Math. Soc.

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“…Assume that the norms of E and F are not order continuous. Then, it follows from the proof of Theorem 1 of Wickstead [14] that E (resp. F) contains a sublattice isomorphic to l ∞ and there exists a positive projection P 1 : E −→ l ∞ (resp.…”

Section: ) the Operator T Is Order Weakly Compact Whenever Its Adjoimentioning

confidence: 95%

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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Converses for the Dodds–Fremlin and Kalton–Saab theorems

Cited by 83 publications

References 7 publications

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

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

Sur les opérateurs de Dunford-Pettis positifs qui sont faiblement compacts

The Duality Problem for the Class of Order Weakly Compact Operators

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