Searcharxiv⌕ Search

arXiv subjects

Antonio Bonilla

Publications and source records attributed to Antonio Bonilla.

11 recordsLinked to original sources
arXiv2026

On the Dynamics of Weighted Composition Operators II

We establish complete characterizations of various notions of expansivity for weighted composition operators on a very general class of locally convex spaces of continuous functions. This class includes several classical classes of continuous function spaces, such as the Banach spaces $C_0(X)$ of continuous scalar-valued functions vanishing at infinity on a Hausdorff locally compact space $X$, endowed with the sup norm, and the locally convex spaces $C(X)_c$ of continuous scalar-valued functions on a completely regular space $X$, endowed with the compact-open topology. We also obtain complete characterizations of various notions of expansivity for weighted composition operators on $L^p(μ)$ spaces, thereby complementing and extending previously known results in the unweighted case. Finally, we establish a conjugation between weighted and unweighted composition operators in the case of dissipative systems on $L^p(μ)$ spaces and apply it to the study of several dynamical properties.

math.DS↗
arXiv2025

On the Dynamics of Weighted Composition Operators

We study the properties of power-boundedness, Li-Yorke chaos, distributional chaos, absolutely Cesàro boundedness and mean Li-Yorke chaos for weighted composition operators on $L^p(μ)$ spaces and on $C_0(Ω)$ spaces. We illustrate the general results by presenting several applications to weighted shifts on the classical sequence spaces $c_0(\mathbb{N})$, $c_0(\mathbb{Z})$, $\ell^p(\mathbb{N})$ and $\ell^p(\mathbb{Z})$ ($1 \leq p < \infty$) and to weighted translation operators on the classical function spaces $C_0[1,\infty)$, $C_0(\mathbb{R})$, $L^p[1,\infty)$ and $L^p(\mathbb{R})$ ($1 \leq p < \infty$).

math.DS↗
arXiv2025

Mean ergodicity of multiplication operators in weighted Dirichlet spaces

We characterize bounded multiplication operators in weighted Dirichlet spaces that are power bounded, Cesàro bounded and uniformly Kreiss. Moreover, we show the equivalence in such spaces between mean ergodicity and Cesàro boundedness for multiplication operators. We perform the same study for adjoints of multiplication operators. As a particular example, we obtain a uniform mean ergodic multiplication operator in Dirichlet spaces that fails to be power bounded.

math.CV↗
arXiv2025

Zero-one law of orbital limit points for weighted shifts

Chan and Seceleanu have shown that if a weighted shift operator on $\ell^p(\mathbb{Z})$, $1\leq p<\infty$, admits an orbit with a non-zero limit point then it is hypercyclic. We present a new proof of this result that allows to extend it to very general sequence spaces. In a similar vein we show that, in many sequence spaces, a weighted shift with a non-zero weakly sequentially recurrent vector has a dense set of such vectors; but an example on $c_0(\mathbb{Z})$ shows that such an operator is not necessarily hypercyclic. On the other hand, we obtain that weakly sequentially hypercyclic weighted shifts are hypercyclic. Chan and Seceleanu have moreover shown that if an adjoint multiplication operator on a Bergman space admits an orbit with a non-zero limit point then it is hypercyclic. We extend this result to very general spaces of analytic functions, including the Hardy spaces.

math.FA↗
arXiv2022

Frequently recurrent operators

Motivated by a recent investigation of Costakis et al. on the notion of recurrence in linear dynamics, we study various stronger forms of recurrence for linear operators, in particular that of frequent recurrence. We study, among other things, the relationship between a type of recurrence and the corresponding notion of hypercyclicity, the influence of power boundedness, and the interplay between recurrence and spectral properties. We obtain, in particular, Ansari- and Léon-Müller-type theorems for $\mathcal{F}$-recurrence under very weak assumptions on the Furstenberg family $\mathcal{F}$. This allows us, as a by-product, to deduce Ansari- and Léon-Müller-type theorems for $\mathcal{F}$-hypercyclicity.

math.FA↗
arXiv2019

Reiterative distributional chaos on Banach spaces

If we change the upper and lower density in the definition of distributional chaos of a continuous linear operator on Banach space by the Banach upper and Banach lower density, respectively, we obtain Li-Yorke chaos. Motivated by this fact, we introduce the notions of reiterative distributional chaos of types $1$, $1^+$ and $2$ for continuous linear operators on Banach spaces, which are characterized in terms of the existence of an irregular vector with additional properties. Moreover, we study its relations with other dynamical properties and give conditions for the existence of a vector subspace $Y$ of $X$ such that every non-zero vector in $Y$ is both irregular for $T$ and distributionally near to zero for $T.$

math.FA↗
arXiv2017

Growth orders and ergodicity for absolutely Cesàro bounded operators

In this paper, we extend the concept of absolutely Cesàro boundedness to the fractional case. We construct a weighted shift operator belonging to this class of operators, and we prove that if $T$ is an absolutely Cesàro bounded operator of order $α$ with $0<α\le 1,$ then $\| T^n\|=o(n^α)$, generalizing the result obtained for $α=1$. Moreover, if $α> 1$, then $\|T^{n}\|= O(n)$. We apply such results to get stability properties for the Cesàro means of bounded operators.

math.FA↗
arXiv2017

Cesàro bounded operators in Banach spaces

We study several notions of boundedness for operators. It is known that any power bounded operator is absolutely Cesàro bounded and strong Kreiss bounded (in particular, uniformly Kreiss bounded). The converses do not hold in general. In this note, we give examples of topologically mixing absolutely Cesàro bounded operators on $\ell^p(\mathbb{N})$, $1\le p < \infty$, which are not power bounded, and provide examples of uniformly Kreiss bounded operators which are not absolutely Cesàro bounded. These results complement very limited number of known examples (see \cite{Shi} and \cite{AS}). In \cite{AS} Aleman and Suciu ask if every uniformly Kreiss bounded operator $T$ on a Banach spaces satisfies that $\lim_n\| \frac{T^n}{n}\|=0$. We solve this question for Hilbert space operators and, moreover, we prove that, if $T$ is absolutely Cesàro bounded on a Banach (Hilbert) space, then $\| T^n\|=o(n)$ ($\| T^n\|=o(n^{\frac{1}{2}})$, respectively). As a consequence, every absolutely Cesàro bounded operator on a reflexive Banach space is mean ergodic, and there exist mixing mean ergodic operators on $\ell^p(\mathbb{N})$, $1< p <\infty$. Finally, we give new examples of weakly ergodic 3-isometries and study numerically hypercyclic $m$-isometries on finite or infinite dimensional Hilbert spaces. In particular, all weakly ergodic strict 3-isometries on a Hilbert space are weakly numerically hypercyclic. Adjoints of unilateral forward weighted shifts which are strict $m$-isometries on $\ell ^2(\mathbb{N})$ are shown to be hypercyclic.

math.FA↗
arXiv2016

Upper frequent hypercyclicity and related notions

Enhancing a recent result of Bayart and Ruzsa we obtain a Birkhoff-type characterization of upper frequently hypercyclic operators and a corresponding Upper Frequent Hypercyclicity Criterion. As an application we characterize upper frequently hypercyclic weighted backward shifts on sequence spaces, which in turn allows us to come up with various counter-examples in linear dynamics that are substantially simpler than those previously obtained in the literature. More generally, we introduce the notion of upper Furstenberg families $\mathcal{A}$ and show that our main results hold for $\mathcal{A}$-hypercyclic operators with respect to such families.

math.FA↗
arXiv2015

Order of growth of distributional irregular entire functions for the differentiation operator

We study the rate of growth of entire functions that are distributionally irregular for the differentiation operator D. More specifically, given $p \in [1,\infty ]$ and $b \in (0,a)$, where $a = \frac{1}{2 max\{2,p\}}$, we prove that there exists a distributionally irregular entire function $f$ for the operator D such that its p-integral mean function $M_p(f,r)$ grows not more rapidly than $e^r r^{-b}$. This completes related known results about the possible rates of growth of such means for D-hypercyclic entire functions. It is also obtained the existence of dense linear submanifolds of H(C) all whose nonzero vectors are D-distributionally irregular and present the same kind of growth.

math.CV↗