On the domination of limited and order Dunford-Pettis operators

H’michane, Jawad; Fahri, Kamal El

doi:10.1007/s40316-015-0036-4

Cited by 6 publications

(3 citation statements)

References 11 publications

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Order By: Relevance

“…Since (x n) is norm bounded in E then it is order bounded, and by the order boundedness of T : E → F it follows that (T (x n )) is order bounded in F . So there exists a positive element y in F such that |T (x n )| ≤ y, and hence from the inequality |f n (T ( x n ))| ≤ |f n | (y) and[13, Lemma 3.1] we see that lim f n (T (x n )) = 0 as desired.…”

mentioning

confidence: 78%

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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mentioning

confidence: 78%

A note on weak almost limited operators

Machrafi¹,

Fahri²,

Moussa³

et al. 2018

HJMS

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“…In this section, we investigate several modifications of limited operators, introduced recently in [16,23,25,26,30,32,33,38]. A bounded subset B ⊆ E ′ is called: c) an almost L-set (shortly, an a-L-set) if every disjoint w-null sequence (x n ) in E is uniformly null on B (cf.…”

Section: Enveloping Normsmentioning

confidence: 99%

“…It is proved in [32] that the domination property is satisfied for: (i) o-limited operators from every E to every Dedekind σ-complete F ;…”

Section: Modifications Of Limited Operatorsmentioning

confidence: 99%

Regularly limited, Grothendieck, and Dunford--Pettis operators

Alpay¹,

Emelyanov²,

Gorokhova³

2022

Preprint

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Various types of compactness in Banach spaces were in streamline of functional analysis in the second half of the 20th century. At some point, in order to diversify the classes of compact operators, the Banach lattice structure had come in play by Meyer-Nieberg, Dodds, Fremlin, Aliprantis, Wickstead, and others. In the present paper, we continue this line of study with main emphasis on regularity of operators under the investigation.

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