arXiv subjects

Luis Alberto Garcia

Publications and source records attributed to Luis Alberto Garcia.

8 recordsLinked to original sources

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.

math.FA↗

arXiv2024

Operators whose adjoints and second adjoints are almost Dunford-Pettis

First we characterize the Banach lattices E whose biduals have the positive Schur property by means of second adjoints of operators on E being almost Dunford-Pettis. Next we extend some known results concerning conditions on the Banach lattices E and F under which the adjoint T* and the second adjoint T** of any positive almost Dunford-Pettis operator T from E to F are almost Dunford-Pettis. Finally, we prove when T* and T** are almost Dunford-Pettis for any (non necessarily almost Dunford-Pettis) T that is either bounded, regular, order bounded or weakly compact.

math.FA↗

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.

math.FA↗

arXiv2022

Aron-Berner extensions of almost Dunford-Pettis multilinear operators

We prove several results establishing conditions on the Banach lattices E_1,..., E_m and F so that the Aron-Berner extensions of (separately) almost Dunford-Pettis m-linear operators from E_1 x ... x E_m to F are (separately) almost Dunford-Pettis. Illustrative examples are provided.

math.FA↗

arXiv2022

Order continuity of Arens extensions of regular multilinear operators

First we give a counterexample showing that recent results on separate order continuity of Arens extensions of multilinear operators cannot be improved to get separate order continuity on the product of the whole of the biduals. Then we establish conditions on the operators and/or on the underlying Riesz spaces/Banach lattices so that the extensions are order continuous on the product of the whole biduals. We also prove that all Arens extensions of any regular multilinear operator are order continuous in at least one variable and we study when Arens extensions of regular homogeneous polynomials on a Banach lattice $E$ are order continuous on $E^{\sim\sim}$.

math.FA↗

arXiv2021

Bidual extensions of Riesz multimorphisms

We prove that all Arens extensions of finite rank Riesz multimorphisms taking values in Archimedean Riesz spaces coincide and are Riesz multimorphisms. Partial results for arbitrary Riesz multimorphisms are obtained. We also prove that, for a class of Banach lattices $F$, which includes $F = c_0, \ell_p, c_0(\ell_p), \ell_p(c_0), \ell_p(\ell_s), 1 < p,s < \infty$, among many others, all Aron-Berner extensions of $F$-valued Riesz multimorphisms between Banach lattices are Riesz multimorphisms.

math.FA↗

arXiv2014

Supercharacters, exponential sums, and the uncertainty principle

The theory of supercharacters, which generalizes classical character theory, was recently introduced by P. Diaconis and I.M. Isaacs, building upon earlier work of C. Andre. We study supercharacter theories on $(Z/nZ)^d$ induced by the actions of certain matrix groups, demonstrating that a variety of exponential sums of interest in number theory (e.g., Gauss, Ramanujan, Heilbronn, and Kloosterman sums) arise in this manner. We develop a generalization of the discrete Fourier transform, in which supercharacters play the role of the Fourier exponential basis. We provide a corresponding uncertainty principle and compute the associated constants in several cases.

math.RT↗

arXiv2013

The graphic nature of the symmetric group

We investigate a remarkable class of exponential sums which are derived from the symmetric groups and which display a diverse array of visually appealing features. Our interest in these expressions stems not only from their astounding visual properties, but also from the fact that they represent a novel and intriguing class of supercharacters.

math.NT↗