arXiv subjects

Arafat Abbar

Publications and source records attributed to Arafat Abbar.

8 recordsLinked to original sources

arXiv2024

Hypercyclic algebras for weighted shifts on trees

We study the existence of algebras of hypercyclic vectors for weighted backward shifts on sequence spaces of directed trees with the coordinatewise product. When $V$ is a rooted directed tree, we show the set of hypercyclic vectors of any backward weighted shift operator on the space $c_0(V)$ or $\ell^1(V)$ is algebrable whenever it is not empty. We provide necessary and sufficient conditions for the existence of these structures on $\ell^p(V), 1<p<+\infty$. Examples of hypercyclic operators not having a hypercyclic algebra are found. We also study the existence of mixing and non-mixing backward weighted shift operators on any rooted directed tree, with or without hypercyclic algebras. The case of unrooted trees is also studied.

math.FA↗

arXiv2024

On several dynamical properties of shifts acting on directed trees

This paper explores the notions of $\mathcal{F}$-transitivity and topological $\mathcal{F}$-recurrence for backward shift operators on weighted $\ell^p$-spaces and $c_0$-spaces on directed trees, where $\mathcal{F}$ represents a Furstenberg family of subsets of $\mathbb{N}_0$. In particular, we establish the equivalence between recurrence and hypercyclicity of these operators on unrooted directed trees. For rooted directed trees, a backward shift operator is hypercyclic if and only if it possesses an orbit of a bounded subset that is weakly dense.

math.FA↗

arXiv2023

A note on the spectrum of Lipschitz operators and composition operators on Lipschitz spaces

Fix a metric space $M$ and let $\mathrm{Lip}_0(M)$ be the Banach space of complex-valued Lipschitz functions defined on $M$. A weighted composition operator on $\mathrm{Lip}_0(M)$ is an operator of the kind $wC_f : g \mapsto w \cdot g \circ f$, where $w : M \to \mathbb C$ and $f: M \to M$ are any map. When such an operator is bounded, it is actually the adjoint operator of a so-called weighted Lipschitz operator $w\widehat{f}$ acting on the Lipschitz-free space $\mathcal F(M)$. In this note, we study the spectrum of such operators, with a special emphasize when they are compact. Notably, we obtain a precise description in the non-weighted $w \equiv 1$ case: the spectrum is finite and made of roots of unity.

math.FA↗

arXiv2023

A pre-adjoint approach on weighted composition operators between spaces of Lipschitz functions

We consider weighted composition operators, that is operators of the type $g \mapsto w \cdot g \circ f$, acting on spaces of Lipschitz functions. Bounded weighted composition operators, as well as some compact weighted composition operators, have been characterized quite recently. In this paper, we provide a different approach involving their pre-adjoint operators, namely the weighted Lipschitz operators acting on Lipschitz free spaces. This angle allows us to improve some results from the literature. Notably, we obtain a distinct characterization of boundedness with a precise estimate of the norm. We also characterise injectivity, surjectivity, compactness and weak compactness in full generality.

math.FA↗

arXiv2023

Orbits of the Backward Shifts with limit points

We show that the bilateral backward shift on $\ell^p(\mathbb{Z},ω)$ that has a projective orbit with a non-zero limit point is supercyclic. This phenomenon holds also for $Γ$-supercyclicity, which extends a result obtained for the first time by Chan and Seceleanu. Moreover, we show that if $K$ is a compact subset of $\ell^p(\mathbb{N},ω)$ such that its orbit under the unilateral backward shift $B$ on $\ell^p(\mathbb{N},ω)$ has a non-zero weak limit point, then $B$ is hypercyclic.

math.FA↗

arXiv2021

Compact and weakly compact Lipschitz operators

Any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended in a unique way to a bounded linear operator $\widehat{f} : \mathcal F(M) \to \mathcal F(N)$ between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for $\widehat{f}$ to be compact in terms of metric conditions on $f$. This extends a result by A. Jiménez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behavior of weakly convergent sequences made of finitely supported elements in Lipschitz-free spaces, we also deduce that $\widehat{f}$ is compact if and only if it is weakly compact.

math.FA↗

arXiv2020

On the dynamics of Lipschitz operators

By the linearization property of Lipschitz-free spaces, any Lipschitz map $f : M \to N$ between two pointed metric spaces may be extended uniquely to a bounded linear operator $\widehat{f} : \mathcal F(M) \to \mathcal F(N)$ between their corresponding Lipschitz-free spaces. In this note, we explore the connections between the dynamics of Lipschitz self-maps $f : M \to M$ and the linear dynamics of their extensions $\widehat{f} : \mathcal F(M) \to \mathcal F(M)$. This not only allows us to relate topological dynamical systems to linear dynamical systems but also provide a new class of hypercyclic operators acting on Lipschitz-free spaces.

math.FA↗

arXiv2020

$Γ$-supercyclicity of families of translates in weighted $L^p$-spaces on locally compact groups

Let $ω$ be a weight function defined on a locally compact group $G$, $1\le p<+\infty$, $S\subset G$ and let us assume that for any $s\in S$, the left translation operator $T_s$ is continuous from the weighted $L^p$-space $L^p(G,ω)$ into itself. For a given set $Γ\subset\mathbb{C}$, a vector $f\in L^p(G,ω)$ is said to be $(Γ,S)$-dense if the set $\{ λT_sf:\, λ\in Γ, \,s\in S\}$ is dense in $L^p(G,ω)$. In this paper, we characterize the existence of $(Γ,S)$-dense vectors in $L^p(G,ω)$ in terms of the weight and the set $Γ$.

math.FA↗