Weak compactness of almost L-weakly and almost M-weakly compact operators

Afkir, Farid; Bouras, Khalid; Elbour, Aziz; Filali, Safae El

doi:10.2989/16073606.2020.1777482

Cited by 6 publications

(5 citation statements)

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“…In [6] it is also constructed numerous distinguished examples of Banach spaces with or without the DP p property. Since the Dunford-Pettis property of order p for Banach spaces and especially Banach lattices was intensively studied by many authors, see for example [1,10,19,20,31]. A more general notion than the DP p property was proposed by Karn and Sinha [26].…”

Section: Definition 12 ([23]mentioning

confidence: 99%

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Dunford–Pettis type properties and the Grothendieck property for function spaces

Gabriyelyan

Ka̧kol

2019

Rev Mat Complut

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For a Tychonoff space X, let C k (X) and Cp(X) be the spaces of real-valued continuous functions C(X) on X endowed with the compact-open topology and the pointwise topology, respectively. If X is compact, the classic result of A. Grothendieck states that C k (X) has the Dunford-Pettis property and the sequential Dunford-Pettis property. We extend Grothendieck's result by showing that C k (X) has both the Dunford-Pettis property and the sequential Dunford-Pettis property if X satisfies one of the following conditions: (i) X is a hemicompact space, (ii) X is a cosmic space (=a continuous image of a separable metrizable space), (iii) X is the ordinal space [0, κ) for some ordinal κ, or (vi) X is a locally compact paracompact space. We show that if X is a cosmic space, then C k (X) has the Grothendieck property if and only if every functionally bounded subset of X is finite. We prove that Cp(X) has the Dunford-Pettis property and the sequential Dunford-Pettis property for every Tychonoff space X, and Cp(X) has the Grothendieck property if and only if every functionally bounded subset of X is finite.2000 Mathematics Subject Classification. Primary 46A03; Secondary 54H11. Key words and phrases. function space, Dunford-Pettis property, Grothendieck property. Jerzy Kakol gratefully acknowledges the nancial support he received from the Center for Advanced Studies in Mathematics of the Ben Gurion University of the Negev during his visit April, 2018.

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Section: Definition 12 ([23]mentioning

confidence: 99%

“…If E is a noremd space, then clearly H is large. If E is a separable metrizable locally convex space, then H is also large by [27, § 29.6 (1)]. Thus H has the DP property (resp., the strict DP p property) by Proposition 5.8.…”

Section: Permanent Propertiesmentioning

confidence: 99%

Dunford–Pettis type properties and the Grothendieck property for function spaces

Gabriyelyan

Ka̧kol

2019

Rev Mat Complut

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“…For example, he proved that L-weakly and M-weakly compact operators are weakly compact (Teorem 5.61 in [3]). Te properties of these classifcations of operators have been investigated and extended to some general cases by some authors; see [5][6][7][8]. In this paper, we introduce an unbounded version for these classifcations of operators as unbounded L-weakly and M-weakly (in short u-Land u-M-weakly) compact operators.…”

Section: Introductionmentioning

confidence: 99%

“…Let y n 􏼈 􏼉 ⊂ Sol((T + S)(U)) be a disjoint sequence, where U is the closed unit ball of X. [5]). Terefore, T + S ∈ LW u (X, E).…”

mentioning

confidence: 99%

Some Properties of Unbounded $M$ -Weakly and Unbounded $L$ -Weakly Compact Operators

Niktab

Azar

Alavizadeh

et al. 2023

Journal of Mathematics

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We introduce the class of unbounded M -weakly compact operators and the class of unbounded L -weakly compact operators. We investigate some properties for this new classification of operators, and we study the relation between them and M -weakly compact and L -weakly compact operators. We also present an operator characterization of Banach lattices with an ordered continuous norm.

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“…Note that an operator T : E ⟶ Y is almost Mweakly compact if and only if its adjoint T ′ is almost Lweakly compact, and an operator T : X ⟶ F is almost Lweakly compact whenever its adjoint T ′ is almost Mweakly compact ([11], Theorem 2.5). The relationship between almost L-weakly (resp., almost M-weakly) compact operators and other classes of operators (e.g., compact operators, weakly compact operators, L-weakly, and M-weakly compact operators) was studied in the literature [11][12][13].…”

Section: Introduction and Notationmentioning

confidence: 99%

On the Lattice Properties of Almost L-Weakly and Almost M-Weakly Compact Operators

Akay

Gök

2021

Journal of Function Spaces

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Weak compactness of almost L-weakly and almost M-weakly compact operators

Cited by 6 publications

References 4 publications

Dunford–Pettis type properties and the Grothendieck property for function spaces

Dunford–Pettis type properties and the Grothendieck property for function spaces

Some Properties of Unbounded $M$ -Weakly and Unbounded $L$ -Weakly Compact Operators

On the Lattice Properties of Almost L-Weakly and Almost M-Weakly Compact Operators

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Weak compactness of almost L-weakly and almost M-weakly compact operators

Cited by 6 publications

References 4 publications

Dunford–Pettis type properties and the Grothendieck property for function spaces

Dunford–Pettis type properties and the Grothendieck property for function spaces

Some Properties of Unbounded M -Weakly and Unbounded L -Weakly Compact Operators

On the Lattice Properties of Almost L-Weakly and Almost M-Weakly Compact Operators

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Some Properties of Unbounded $M$ -Weakly and Unbounded $L$ -Weakly Compact Operators