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<p>In this paper, we introduce and study some new classes of sets and operators known as strong weak limited sets, almost <i>L</i>-Grothendieck sets, strong weak limited operators and almost <i>L</i>-Grothendieck operators. We establish some characterizations of the two classes of sets and operators. After that, we present some relationships between strong weak limited (resp. almost <i>L</i>- Grothendieck) operators and those whose adjoint maps almost <i>L</i>-sets into relatively (resp. weakly) compact ones. In the sequel, we realize that strong weak limited operators extend discrete dual Banach lattices with order continuous norm, in the sense that the identity operator of a Banach lattice is strong weak limited, if and only if, its dual is discrete with order continuous norm, which enables us to extract a new characterization of discrete dual Banach lattices with order continuous norms. Furthermore, we characterize weak Dunford-Pettis* property through strong weak limited operators. Besides, we study the domination property of the class of strong weak limited operators. Finally, we present some connections between almost Grothendieck, almost <i>L</i>-Grothendieck and Grothendieck operators.</p>

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Strong weak limited and almost L-Grothendieck sets and operators

  • 期刊论文

  • 发表日期：

    2025年2月15日

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  • Farid Afkir,
  • Fatima Zahra Oughajji

摘要

In this paper, we introduce and study some new classes of sets and operators known as strong weak limited sets, almost L-Grothendieck sets, strong weak limited operators and almost L-Grothendieck operators. We establish some characterizations of the two classes of sets and operators. After that, we present some relationships between strong weak limited (resp. almost L- Grothendieck) operators and those whose adjoint maps almost L-sets into relatively (resp. weakly) compact ones. In the sequel, we realize that strong weak limited operators extend discrete dual Banach lattices with order continuous norm, in the sense that the identity operator of a Banach lattice is strong weak limited, if and only if, its dual is discrete with order continuous norm, which enables us to extract a new characterization of discrete dual Banach lattices with order continuous norms. Furthermore, we characterize weak Dunford-Pettis* property through strong weak limited operators. Besides, we study the domination property of the class of strong weak limited operators. Finally, we present some connections between almost Grothendieck, almost L-Grothendieck and Grothendieck operators.