arXiv subjects

Vinícius C. C. Miranda

Publications and source records attributed to Vinícius C. C. Miranda.

8 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.

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arXiv2026

Arens extensions of disjointness preserving multilinear operators

First we give a counterexample showing that, unlike the second adjoint of a disjointness preserving linear operator, Arens extensions of disjointness preserving multilinear operators are not disjointness preserving in general. Then we prove that, if $E_1, \ldots, E_m$ are Riesz spaces with $F$ Archimedean and $A \colon E_1 \times \cdots \times E_m \to F$ is a regular disjointness preserving $m$-linear operator, then all Arens extensions of $A$ are disjointness preserving if: (i) $A$ has finite lattice rank; or (ii) $E_1, \ldots, E_m$ are also Archimedean and the extensions are restricted to the product of the order continuous biduals; or (iii) the spaces are Banach lattices and either $E_1^*, \ldots, E_m^*$ have order continuous norms or $F^*$ has a Schauder basis consisting of disjointness preserving functionals.

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arXiv2026

A $p$-summability approach to the weak maximizing property

Motivated by the weak maximizing property ($\mathrm{WMP}$), we investigate $p$-summability conditions on maximizing sequences of bounded linear operators. Since the naive $p$-summability formulation is independent of $p$ and collapses to norm attainment for all operators, we introduce the weakly $p$-singular maximizing property ($\mathrm{SMP}_p$), based on maximizing sequences with no weakly $p$-summable subsequence. We completely characterize when the pairs $(\ell_p,\ell_q)$ and $(\ell_p,c_0)$ have the $\mathrm{SMP}_r$, revealing sharp contrasts with the $\mathrm{WMP}$. We also characterize the Schur property of order $p$ via the $\mathrm{WMP}$ and the $\mathrm{SMP}_r$. Under relative weak $p$-precompactness and suitable Dunford-Pettis-type assumptions, we characterize universal norm attainment for operators from $X$ to $Y$ in terms of either the $\mathrm{SMP}_q$ for adjoint operators or the weak$^{*}$-to-weak$^{*}$ maximizing property, and derive corresponding duality consequences for the $\mathrm{SMP}_p$. Finally, we introduce $p$-convergent perturbation properties for operators and their adjoints, characterize them for classical sequence spaces, and show that the $p$-convergent perturbation property is strictly weaker than the $\mathrm{SMP}_p$.

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arXiv2025

Gelfand-Phillips-type properties in Banach lattices

We study $p$-limited and almost $p$-limited sets in Banach lattices and their connections with relatively $p$-compact and relatively compact sets. We investigate the weak and the strong Gelfand-Phillips property of order $p$, as well as the $p$-GP property introduced by Delgado and Piñeiro, providing conditions under which these properties may coincide. Additionally, we prove that a Banach lattice $E$ is a KB-space if and only if every almost $p$-limited set in $E$ is relatively weakly compact if and only if every the adjoint of a weakly compact taking values on $E$ is disjoint $p$-summing.

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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.

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arXiv2024

Banach lattices of homogeneous polynomials not containing $c_0$

First we develop a technique to construct Banach lattices of homogeneous polynomials. We obtain, in particular, conditions for the linear spans of all positive compact and weakly compact $n$-homogeneous polynomials between the Banach lattices $E$ and $F$, denoted by ${\cal P}_{\cal K}^r(^n E; F)$ and $\mathcal{P}_{\mathcal{W}}^r(^n E; F)$, to be Banach lattices with the polynomial regular norm. Next we study when the following are equivalent for ${\cal I} = {\cal K}$ or ${\cal I} = {\cal W}$: (1) The space $\mathcal{P}^r(^n E; F)$ of regular polynomials contains no copy of $c_0$. (2) ${\cal P}_{\mathcal{I}}^r(^n E; F)$ contains no copy of $c_0$. (3) ${\cal P}_{\mathcal{I}}^r(^n E; F)$ is a projection band in $\mathcal{P}^r(^n E; F)$. (4) Every positive polynomial in $\mathcal{P}^r(^n E; F)$ belongs to ${\cal P}_{\cal I}^r(^nE;F)$. The result we obtain in the compact case can be regarded as a lattice polynomial Kalton theorem. Most of our results and examples are new even in the linear case $n = 1$.

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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 $DP^*P_p$.

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arXiv2023

Disjoint $p$-convergent operators and their adjoints

First we give conditions on a Banach lattice $E$ so that an operator $T$ from $E$ to any Banach space is disjoint $p$-convergent if and only if $T$ is almost Dunford-Pettis. Then we study when adjoints of positive operators between Banach lattices are disjoint $p$-convergent. For instance, we prove that the following conditions are equivalent for all Banach lattices $E$ and $F$: (i) A positive operator $T \colon E \to F$ is almost weak $p$-convergent if and only if $T^*$ is disjoint $p$-convergent; (ii) $E^*$ has order continuous norm or $F^*$ has the positive Schur property of order $p$. Very recent results are improved, examples are given and applications of the main results are provided.

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