arXiv subjects

José Lucas P. Luiz

Publications and source records attributed to José Lucas P. Luiz.

7 recordsLinked to original sources

arXiv2026

Norm attainment for multilinear operators and polynomials on Banach Spaces and Banach lattices

We study norm attainment for multilinear operators and homogeneous polynomials between Banach spaces, as well as for positive multilinear operators between Banach lattices. We establish multilinear and polynomial versions of [23, Theorem B] and [35, Theorem 2.12]. More precisely, we provide sufficient conditions on Banach spaces $X_1, \dots, X_n$ and $Y$ ensuring that every $A \in \mathcal{L}(X_1, \dots, X_n; Y)$ (respectively, $P \in \mathcal{P}(^n X_1; Y)$) is weakly sequentially continuous if and only if it attains its norm. We also obtain analogous results for positive $n$-linear operators and positive $n$-homogeneous polynomials in the setting of Banach lattices.

math.FA↗

arXiv2025

On norm-attaining positive operators between Banach lattices

In this paper we study the norm-attainment of positive operators between Banach lattices. By considering an absolute version of James boundaries, we prove that: If $E$ is a reflexive Banach lattice whose order is given by a basis and $F$ is a Dedekind complete Banach lattice, then every positive operator from $E$ to $F$ is compact if and only if every positive operator from $E$ to $F$ attains its norm. An analogue result considering that $E$ is reflexive and the order in $F$ is continuous and given by a basis was proven. We applied our result to study a positive version of the weak maximizing property.

math.FA↗

arXiv2023

Disjoint Dunford-Pettis-type properties in Banach lattices

New characterizations of the disjoint Dunford-Pettis property of order $p$ (disjoint $DPP_p$) are proved and applied to show that a Banach lattice of cotype $p$ has the disjoint $DPP_p$ 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 $DPP_p$, 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 $DPP_p$ and of the disjoint $DP^*P_p$.

math.FA↗

arXiv2023

Grothendieck's compactness principle for the absolute weak topology

We prove the following results: (i) Every absolutely weakly compact set in a Banach lattice is absolutely weakly sequentially compact. (ii) The converse of (i) holds if $E$ is separable or $B_{E^{**}}$ is absolutely weak$^*$ compact. (iii) Every absolutely weakly compact subset of a Banach lattice is contained in the closed convex hull of an absolutely weakly null sequence if and only if the Banach lattice has the positive Schur property. Examples and applications are provided.

math.FA↗

arXiv2020

Complete latticeability in vector-valued sequence spaces

First we adjust a technique due to Jiménez-Rodríguez to prove the complete latticeability of the set of disjoint non-norm null weakly null sequences and of the set of disjoint non-norm null regular-polynomially null sequences in Banach lattices. Then we apply the mother vector technique to prove the complete latticeability of $λ_π(E) \setminus λ_s(E)$, which implies the complete latticeability of $(\ell_p\widehat{\otimes}_{|π|}E)\setminus(\ell_p\widehat{\otimes}_πE)$, where $E$ is a Banach lattice and $1 < p < \infty$.

math.FA↗

arXiv2020

The positive polynomial Schur property in Banach lattices

We study the class of Banach lattices that are positively polynomially Schur. Plenty of examples and counterexamples are provided, lattice properties of this class are proved, arbitrary $L_p(μ)$-spaces are shown to be positively polynomially Schur, lattice analogues of results on Banach spaces are obtained and relationships with the positive Schur and the weak Dunford-Pettis properties are established.

math.FA↗

arXiv2019

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

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.

math.FA↗