The Property (D) and the Almost Limited Completely Continuous Operators

Lourenço, Mary Lilian; Miranda, V. C. C.

doi:10.1007/s10476-023-0190-x

Cited by 3 publications

(1 citation statement)

References 15 publications

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“…In the recent paper [4], H. Ardakani and K. Amjadi introduced the positive DP * property of order p considering the positively limited sets introduced in [5]. It was observed before [4,Theorem 3.6] that a Banach lattice E with the property (d) (see [27] for details) has the positive DP * property of order p if and only if E has the p-weak DP * . Since σ-Dedekind complete Banach lattices have the property (d), the following is a straightforward consequence of Theorem 3.7: Corollary 3.8.…”

Section: Introductionmentioning

confidence: 99%

On the Schur, positive Schur and weak Dunford–Pettis properties in Fréchet lattices

Botelho

Luiz

2021

Ann. Fenn. Math.

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We prove some general results on sequential convergence in Fréchet lattices that yield, as particular instances, the following results regarding a closed ideal I of a Banach lattice E: (i) If two of the lattices E, I and E/I have the positive Schur property (the Schur property, respectively) then the third lattice has the positive Schur property (the Schur property, respectively) as well; (ii) If I and E/I have the dual positive Schur property, then E also has this property; (iii) If I has the weak Dunford-Pettis property and E/I has the positive Schur property, then E has the weak Dunford-Pettis property. Examples and applications are provided.Definition 1.1. A property P of Banach lattices is a 3-lattice property if the following holds: given a closed ideal I of the Banach lattice E, if two of the lattices E, I and E/I have P, then the third one has P as well.A Banach lattice has the positive Schur property if weakly null sequences formed by positive vectors are norm null. A lot of research has been done on this property, for some recent contributions see, e.g., [6,7,13,18,24,25,26,28]. Among other results, in this paper we prove that, contrary to the case of the Schur property for Banach spaces, the Schur and the positive Schur properties are 3-lattice properties. These results will appear as applications of general results on the Schur and the positive Schur properties in Fréchet lattices.

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Section: Introductionmentioning

confidence: 99%

On the Schur, positive Schur and weak Dunford–Pettis properties in Fréchet lattices

Botelho

Luiz

2021

Ann. Fenn. Math.

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Some properties of 𝑝-limited sets

Galindo¹,

Miranda²

2023

Proc. Amer. Math. Soc.

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Karn and Sinha [Glasg. Math. J. 56 (2014), pp. 427–437] introduced the p p -limited ( 1 ≤ p > ∞ 1 \leq p > \infty ) sets (see the definition below). We show that p p -limited sets are preserved by continuous polynomials as well as by the projective tensor product and that scalar-valued polynomials are p p -summable on p p -limited sets. Considering the notion of p p -limited set from the ℓ p \ell _p -valued operators point of view, we introduce in Section \ref{sec:3} two weaker types of p p -limitedness in the setting of Banach lattices and study their basic properties.

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Disjoint Dunford–Pettis-Type Properties in Banach Lattices

Botelho,

Luiz,

Miranda

2024

The Quarterly Journal of Mathematics

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New characterizations of the disjoint Dunford–Pettis property of order p (disjoint DPPp) are proved and applied to show that a Banach lattice of cotype p has the disjoint DPPp whenever its dual has this property. The disjoint Dunford–Pettis$^*$ property of order p (disjoint $DP^*P_p$) is thoroughly investigated. Close connections with the positive Schur property of order p, with the disjoint DPPp, with the p-weak $DP^*$ property and with the positive $DP^*$ property of order p are established. In a final section, we study the polynomial versions of the disjoint DPPp and of the disjoint $DP^*P_p$.

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The Property (D) and the Almost Limited Completely Continuous Operators

Cited by 3 publications

References 15 publications

On the Schur, positive Schur and weak Dunford–Pettis properties in Fréchet lattices

On the Schur, positive Schur and weak Dunford–Pettis properties in Fréchet lattices

Some properties of 𝑝-limited sets

Disjoint Dunford–Pettis-Type Properties in Banach Lattices

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