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Mohamed Amouch

Publications and source records attributed to Mohamed Amouch.

16 recordsLinked to original sources
arXiv2023

Hypercyclicity of operators that $λ$-commute with the Hardy backward shift

An operator $T$ acting on a separable complex Hilbert space $H$ is said to be hypercyclic if there exists $f\in H$ such that the orbit $\{T^n f:\ n\in \mathbb{N}\}$ is dense in $H$. Godefroy and Shapiro \cite{GoSha} characterized those elements in the commutant of the Hardy backward shift which are hypercyclic. In this paper we study some dynamics properties of operators $X$ that $λ$-commute with the Hardy backward shift $B$, that is, $BX=λXB$.

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.

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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.

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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↗
arXiv2019

Supercyclicity of the left and right multiplication operators on Banach ideal of operators

Let $X$ be a Banach space with $\dim X>1$ such that $X^{\ast}$, its dual, is separable and $\mathcal{B}(X)$ the algebra of bounded linear operators on $X$. In this paper, we study the passage of property of being supercyclic from an operator $T\in\mathcal{B}(X)$ to the left and right multiplication induced by $T$ on separable admissible Banach ideal of $\mathcal{B}(X)$. We give a sufficient condition for the tensor product $T\widehat{\otimes}R$ of two operators to be supercyclic. As a consequence, we give another equivalent conditions for the Supercyclicity Criterion.

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↗
arXiv2017

Spectra Originated from Fredholm Theory and Browder's Theorem

We give a new characterization of Browders theorem through equality between the pseudo B-Weyl spectrum and the generalized Drazin spectrum. Also, we will give conditions under which pseudo B-Fredholm and pseudo B-Weyl spectrum introduced in [11] and [30] become stable under commuting Riesz perturbations.

math.SP↗
arXiv2016

Pseudo B-Fredholm Operators and Spectral Theory

In this paper, we show that every pseudo B-Fredholm operator is a pseudo Fredholm operator. Afterwards, we prove that the pseudo B-Weyl spectrum is empty if and only if the pseudo B-Fredholm spectrum is empty. Also, we study a symmetric difference between some parts of the spectrum.

math.SP↗
arXiv2014

Some spectral properties for generalized derivations

Given Banach spaces $\mathcal{X}$ and $\mathcal{Y}$ and Banach space operators $A\in L(\mathcal{X})$ and $B\in L(\mathcal{Y}).$ The generalized derivation $δ_{A,B} \in L(L(\mathcal{Y},\mathcal{X}))$ is defined by $δ_{A,B}(X)=(L_{A}-R_{B})(X)=AX-XB$. This paper is concerned with the problem of the transferring the left polaroid property, from operators $A$ and $B^{*}$ to the generalized derivation $δ_{A,B}$. As a consequence, we give necessary and sufficient conditions for $δ_{A,B}$ to satisfy generalized a-Browder's theorem and generalized a-Weyl's theorem. As application, we extend some recent results concerning Weyl type theorems.

math.SP↗
arXiv2013

Generalized Browder's and Weyl's Theorems for Generalized Derivations

Given Banach spaces $\X$ and $\Y$ and Banach space operators $A\in L(\X)$ and $B\in L(\Y)$, let $ρ\colon L(\Y,\X)\to L(\Y,\X)$ denote the generalized derivation defined by $A$ and $B$, i.e., $ρ(U)=AU-UB$ ($U\in L(\Y,\X)$). The main objective of this article is to study Weyl and Browder type theorems for $ρ\in L(L(\Y,\X))$. To this end, however, first the isolated points of the spectrum and the Drazin spectrum of $ρ\in L(L(\Y,\X))$ need to be characterized. In addition, it will be also proved that if $A$ and $B$ are polaroid (respectively isoloid), then $ρ$ is polaroid (respectively isoloid).

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