The class of b-limited operators

Kaddouri, A. El; Fahri, Kamal El; Moussa, Mohammed

doi:10.14232/actasm-014-570-2

Cited by 7 publications

(6 citation statements)

References 6 publications

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“…Recently, some order type operators were introduced and studied. An operator T : E → X is said to be order limited [12], if T carries each order bounded subset of E to a limited one in X. The dual counterpart of an order limited operator is defined in [10] as follows: an operator T : X → E is called order (L)-Dunford-Pettis, if the adjoint T ′ carries each order bounded subset of E ′ to an (L)-set in X ′ .…”

Section: The Product Of Wa-limited Operators By Some Order Type Operatorsmentioning

confidence: 99%

“…The dual counterpart of an order limited operator is defined in [10] as follows: an operator T : X → E is called order (L)-Dunford-Pettis, if the adjoint T ′ carries each order bounded subset of E ′ to an (L)-set in X ′ . The following two sequential characterizations were established for the two latter types of operators (see [12,Theorem 3.3] and [10, Theorem 2.5]).…”

Section: The Product Of Wa-limited Operators By Some Order Type Operatorsmentioning

confidence: 99%

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A note on weak almost limited operators

Machrafi¹,

Fahri²,

Moussa³

et al. 2018

HJMS

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Let us recall that an operator T : E → F, between two Banach lattices, is said to be weak* Dunford-Pettis (resp. weak almost limited) if fn (T xn) → 0 whenever (xn) converges weakly to 0 in E and (fn) converges weak* to 0 in F (resp. fn (T xn) → 0 for all weakly null sequences (xn) ⊂ E and all weak* null sequences (fn) ⊂ F with pairwise disjoint terms). In this note, we state some sufficient conditions for an operator R : G → E(resp. S : F → G), between Banach lattices, under which the product T R (resp. ST) is weak* Dunford-Pettis whenever T : E → F is an order bounded weak almost limited operator. As a consequence, we establish the coincidence of the above two classes of operators on order bounded operators, under a suitable lattice operations' sequential continuity of the spaces (resp. their duals) between which the operators are defined. We also look at the order structure of the vector space of weak almost limited operators between Banach lattices.

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Section: The Product Of Wa-limited Operators By Some Order Type Operatorsmentioning

confidence: 99%

Section: The Product Of Wa-limited Operators By Some Order Type Operatorsmentioning

confidence: 99%

A note on weak almost limited operators

Machrafi¹,

Fahri²,

Moussa³

et al. 2018

HJMS

Self Cite

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“…Alternatively, an operator T : X −→ Y is limited if, and only if, T ( f n ) −→ 0 for every weak* null sequence ( f n ) ⊂ Y . Following [8], an operator T from a Banach lattice E into X is said to be order limited if it carries each order bounded subset of E into a limited set of X . This class of operators is bigger than the class of limited operators.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

On the domination of limited and order Dunford-Pettis operators

H’michane

Fahri

2015

Ann. Math. Québec

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We study the domination problem for the class of limited operators and that of order limited operators. On the other hand, we show that the class of order Dunford-Pettis operators satisfies the domination problem.Keywords Order-Dunford-Pettis operator · Limited operator · Order limited operator · Order continuous norm · Weak* sequentially continuous lattice operations · Dedekind σ -complete Banach lattice Résumé Nous étudions le problème de domination pour la classe des opérateurs limités et celle des opérateurs limités pour l'ordre. D'autre part, nous montrons que la classe des opérateurs de Dunford-Pettis pour l'ordre satisfait le problème de domination. Mathematics Subject Classification

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“…Note that a Banach space X has the Dunford-Pettis (respectively, Dunford-Pettis*) property if f n (x n ) → 0 for every weakly null sequence (x n ) ⊂ X and every weakly (respectively, weak*) null sequence (f n ) ⊂ X ′ . Finally, we remember that an operator T : E → X is: ⊲ order weakly compact, if the image by T of each order bounded subset of E is a relatively weakly compact subset of X; ⊲ order limited, if T carries each order bounded subset in E to a limited one in X, equivalently, [5], Theorem 3.3 (3); ⊲ AM-compact, if the image by T of each order bounded subset of E is a relatively compact subset of X.…”

Section: Introduction and Notationmentioning

confidence: 99%

Application of~$\rm(L)$~sets to some classes of operators

Fahri¹,

Machrafi²,

H’michane³

et al. 2016

Math.Boh.

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Application of (L) sets to some classes of operators Mathematica Bohemica, Vol. 141 (2016) Abstract. The paper contains some applications of the notion of (L) 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 → E from a Banach space into a Banach lattice is order (L)-Dunford-Pettis, if and only if |T (xn)| → 0 for σ(E, E ′ ) for every weakly null sequence (xn) ⊂ X. We also investigate relationships between order (L)-DunfordPettis, AM-compact, weak* Dunford-Pettis, and Dunford-Pettis operators. In particular, it is established that each operator T : E → F between Banach lattices is Dunford-Pettis whenever it is both order (L)-Dunford-Pettis and weak* Dunford-Pettis, if and only if E has the Schur property or the norm of F is order continuous.

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The class of b-limited operators

Cited by 7 publications

References 6 publications

A note on weak almost limited operators

A note on weak almost limited operators

On the domination of limited and order Dunford-Pettis operators

Application of~$\rm(L)$~sets to some classes of operators

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