arXiv subjects

Otmane Benchiheb

Publications and source records attributed to Otmane Benchiheb.

12 recordsLinked to original sources

arXiv2026

On Topological Numerical Transitivity, C*-Transitivity and generalizations

Motivated by the concept of numerical hypercyclicity, in this paper, we introduce three new notions in linear dynamics: topological numerical transitivity, generalized numerical transitivity, C*-transitivity. The first notion is defined for operators on general Banach spaces, whereas the latter two are formulated in the setting of operators on Banach algebras. The concept of C*-transitivity is also applicable to operators on the space of Hilbert-Schmidt operators. We prove that these new notions are mutually different, and we also show that they differ from the standard notions in dynamics. In particular, while topological numerical transitivity implies numerical hypercyclicity, as we argue in the paper, we provide examples of numerically hypercyclic and strongly numerically hypercyclic operators that are not topologically numerically transitive. Also, we construct nonsupercyclic C* transitive and generalized numerically transitive operators on the C*-algebra of compact operators on a separable Hilbert space, as well as C*-transitive nonsupercyclic operators on the standard Hilbert C*-module. In addition, we study diagonal operators on finite-dimensional and separable Hilbert spaces, and we obtain complete characterizations of both numerical hypercyclicity and topological numerical transitivity in terms of coefficient-simplex criteria. Further, we provide some sufficient conditions for diagonal operators on the standard Hilbert C*-module to be C*-transitive. All these results are illustrated with concrete examples. At the end of the paper, we compare topological numerical transitivity and C*-transitivity with numerous standard notions in linear dynamics, and we prove that topological numerical transitivity and C*-transitivity genuinely differ from all these standard concepts in dynamics.

math.FA↗

arXiv2026

A Metric Framework for Approximate Transitivity, Mixing, and Hypercyclicity

We study metric versions of transitivity, mixing, and hypercyclicity for continuous maps, based on intersections of the form $ f^{n}(U)\cap B_δ(V)\neq\varnothing. $ We introduce $δ$-topological transitivity, $δ$-topological mixing, and a uniform-from-below version of $δ$-mixing, and prove $ \mathrm{UFB\mbox{-}}δ\text{-TM} \;\Rightarrow\; δ\text{-TM} \;\Rightarrow\; δ\text{-TT}. $ In the linear setting of separable F-spaces, we formulate a $δ$-Hypercyclicity Criterion, prove that it implies $δ$-hypercyclicity, and show that the classical Hypercyclicity Criterion implies the $δ$-criterion for every $δ>0$. We further show that this criterion yields eventual $δ$-mixing along the underlying sequence. Finally, we discuss weighted backward shifts, derive sufficient conditions for $δ$-topological mixing, and show that $λB$ satisfies the $δ$-Hypercyclicity Criterion for every $δ>0$.

math.FA↗

arXiv2025

Luh hypercyclic vector for composition operator

In this paper, we deal with the construction of holomorphic functions on a simply connected domain satisfying that all its derivatives and antiderivatives under a composition operator have a dense orbit. Such functions will be called Luh hypercyclic vectors for the respective composition operator. We show that there is a dense linear manifold of Luh hypercyclic vectors. Moreover, we study the dynamics of cosine operator function generated by weighted composition operators on solid Banach function spaces, in particular on Orlicz and Morrey spaces, and we give sufficient conditions for supercyclicity of such cosine operator functions in terms of the corresponding weight function. Also, we give concrete examples of weighted translations satisfying these sufficient conditions.

math.FA↗

arXiv2022

Disjoint strong transitivity of composition operators

A Furstenberg family $\mathcal{F}$ is a collection of infinite subsets of the set of positive integers such that if $A\subset B$ and $A\in \mathcal{F}$, then $B\in \mathcal{F}$. For a Furstenberg family $\mathcal{F}$, finitely many operators $T_1,...,T_N$ acting on a common topological vector space $X$ are said to be disjoint $\mathcal{F}$-transitive if for every non-empty open subsets $U_0,...,U_N$ of $X$ the set $\{n\in \mathbb{N}:\ U_0 \cap T_1^{-n}(U_1)\cap...\cap T_N^{-n}(U_N)\neq\emptyset\}$ belongs to $\mathcal{F}$. In this paper, depending on the topological properties of $Ω$, we characterize the disjoint $\mathcal{F}$-transitivity of $N\geq2$ composition operators $C_{ϕ_1},\ldots,C_{ϕ_N}$ acting on the space $H(Ω)$ of holomorphic maps on a domain $Ω\subset \mathbb{C}$ by establishing a necessary and sufficient condition in terms of their symbols $ϕ_1,...,ϕ_N$.

math.FA↗

arXiv2022

On recurrent sets of operators

An operator $T$ acting on a Banach space $X$ is said to be recurrent if for each $U$; a nonempty open subset of $X$, there exists $n\in\mathbb{N}$ such that $T^n(U)\cap U\neq\emptyset.$ In the present work, we generalize this notion from a single operator to a set $Γ$ of operators. As application, we study the recurrence of $C$-regularized group of operators.

math.FA↗

arXiv2021

Recurrence of multiples of composition operators on weighted Dirichlet spaces

A bounded linear operator $T$ acting on a Hilbert space $\mathcal{H}$ is said to be recurrent if for every non-empty open subset $U\subset \mathcal{H}$ there is an integer $n$ such that $T^n (U)\cap U\neq\emptyset$. In this paper, we completely characterize the recurrence of scalar multiples of composition operators, induced by linear fractional self maps of the unit disk, acting on weighted Dirichlet spaces $S_ν$; in particular on the Bergman space, the Hardy space, and the Dirichlet space. Consequently, we complete a previous work of Costakis et al. \cite{costakis} on recurrence of linear fractional composition operators on Hardy space. In this manner, we determine the triples $(λ,ν,ϕ)\in \mathbb{C}\times \mathbb{R}\times LFM(\mathbb{D})$ for which the scalar multiple of composition operator $λC_ϕ$ acting on $S_ν$ fails to be recurrent.

math.FA↗

arXiv2021

On super-rigid and uniformly super-rigid operators

An operator $T$ acting on a Banach space $X$ is said to be super-recurrent if for each open subset $U$ of $X$, there exist $λ\in\mathbb{K}$ and $n\in \mathbb{N}$ such that $λT^n(U)\cap U\neq\emptyset$. In this paper, we introduce and study the notions of super-rigidity and uniform super-rigidity which are related to the notion of super-recurrence. We investigate some properties of these classes of operators and show that they share some properties with super-recurrent operators. At the end, we study the case of finite-dimensional spaces.

math.FA↗

arXiv2021

On super-recurrent operators

In this paper, we introduce and study the notion of super-recurrence of operators. We investigate some properties of this class of operators and show that it shares some characteristics with supercyclic and recurrent operators. In particular, we show that if $T$ is super-recurrent, then $σ(T)$ and $σ_p(T^*)$, the spectrum of $T$ and the point spectrum of $T^*$ respectively, have some noteworthy properties.

math.FA↗

arXiv2021

Codiskcyclic sets of operators on complex topological vector spaces

Let $X$ be a complex topological vector space and $L(X)$ the set of all continuous linear operators on $X.$ In this paper, we extend the notion of the codiskcyclicity of a single operator $T\in L(X)$ to a set of operators $Γ\subset L(X).$ We prove some results for codiskcyclic sets of operators and we establish a codiskcyclicity criterion. As an application, we study the codiskcyclicity of $C_0$-semigroups of operators.

math.FA↗

arXiv2019

Diskcyclicity of sets of operators and applications

In this paper, we extend the notion of diskcyclicity and disk transitivity of a single operator to a subset of $\mathcal{B}(X)$. We establish a diskcyclicity criterion and we give the relationship between this criterion and the diskcyclicity. As applications, we study the diskcyclicty of $C_0$-semigroups and $C$-regularized groups of operators. We show that a diskcyclic $C_0$-semigroup exists on a complex topological vector space $X$ if and only if dim$(X)=1$ or dim$(X)=\infty$ and we prove that diskcyclicity and disk transitivity of a $C_0$-semigroups and $C$-regularized groups are equivalent.

math.FA↗

arXiv2018

On Linear Dynamics of Sets of Operators

Let $X$ be a complex topological vector space with $dim(X)>1$ and $\mathcal{B}(X)$ the set of all continuous linear operators on $X$. The concept of hypercyclicity for a subset of $\mathcal{B}(X)$, was introduced in \cite{AKH}. In this work, we introduce the notion of hypercyclic criterion for a subset of $\mathcal{B}(X)$. We extend some results known for a single operator and $C_0$-semigroup to a subset of $\mathcal{B}(X)$ and we give applications for $C$-regularized groups of operators.

math.DS↗

arXiv2018

On supercyclic sets of operators

Let $X$ be a complex topological vector space with dim$(X)>1$ and $\mathcal{B}(X)$ the space of all continuous linear operators on $X$. In this paper, we extend the concept of supercyclicity of a single operators and strongly continuous semigroups of operators to a subset of $\mathcal{B}(X)$. We establish some results for supercyclic set of operators and we give some applications for strongly continuous semigroups of operators and $C$-regularized group of operators.

math.FA↗