Mohamed Amouch

• Full Professor
• Director of the Laboratory Fundamental Mathematics and their Applications at University Chouaïb Doukkali

Let T be a bounded linear operator acting on a Banach space X such that T or T* has the SVEP. We prove that the spectral mapping theorem holds for the semi-essential approximate point spectrum ¿SBF-+ (T); and we show that generalized a-Browder's theorem holds for f(T) for every analytic function f defined on an open neighbourhood U of [sigma](T): M...

. In this note we introduce and study the property (gw), which extends property (w) introduced by Rakoc̆evic in [23]. We investigate the property (gw) in connection with Weyl type theorems. We show that if T is a bounded linear operator T acting on a Banach space X, then property (gw) holds for T if and only if property (w) holds for T and Π a (T)...

A Banach space operator $T$ satisfies generalized a-Weyl's theorem if the complement of its upper semi B-Weyl spectrum in its approximate point spectrum is the set of eigenvalues of $T$ which are isolated in the approximate spectrum of $T.$ In this note we characterize hypecyclic and supercyclic operators satisfying generalized a-Weyl's theorem.

Dans ce travail, on étudie le quasi-adjoint d’une bi-multiplication sur l’algèbre des opérateurs linéaires bornés sur un espace de Hilbert. Nous caractérisons la positivité d’une bi-multiplication. Ce qui nous permet d’établir une décomposition polaire pour cette classe d’opérateurs élémentaires. Par le biais du quast-adjoint d’une bi-multiplicatio...

In this paper, we introduce the notions of hypercyclicity and transitivity for random dynamical systems and we study specific properties related to these concepts. Additionally, we establish connections between them and irreducible Markov chains.

In this paper, we consider a generalization of $(A, m, n)$-isosymmetric operators on a Hilbert space to the multivariable setting. For a $d$-tuple of commuting operators $\mathbf{T}=(T_1 , \dots, T_d)$, $A$ a positive operator and $m, n$ are two positive integers, we introduce the class of $(A, m, n)$-isosymmetric $d$-tuple of commuting operators a...

An operator T acting on a metric compact space X is said to be recurrent if, for each nonempty open subset U of X, there exists n ∈ N such that T n (U) ∩ U ̸ = ∅. In this paper, we introduce and study the notion of recurrence of interval map. We investigate some properties of this class of operators and show the links between recurrence, R-mixing a...

This paper examines the phenomenon of super-recurrence exhibited by multiplication operators when applied to various function spaces. Our findings demonstrate that, contrary to hypercyclicity and su-percyclicity cases, certain multiplication operators on specific function spaces exhibit super-recurrent behavior.

In this paper we introduce and study the notion of Cesàro-recurrent operators. We give some necessary and sufficient conditions for an operator T acting on a Banach space X to be Cesàro-recurrent.

In this article, we introduce and study the notion of disjoint topologically super-recurrent operators for finitely many operators acting on a complex Banach space. As an application, we characterize the disjoint topological super-recurrence of finitely many different powers of unilateral and bilateral weighted shifts.

In this paper, we extend the concepts of cyclic transitivity, recurrence, and super-recurrence by introducing and examining a novel notion called C-recurrence. We then present a C-recurrence Criterion, inspired by the Hypercyclicity Criterion and the Cyclicity Criterion. Finally, we characterize C-recurrence for weighted shifts.

In this research, the Ω-hypercyclic and Ω-transitive behavior are studied within the framework of linear strongly continuous semigroups. We give sufficient constraints on the spectrum of an operator to yield a Ω-hypercyclic semigroup. Also, we establish necessary and sufficient conditions on the semigroup to be Ω-transitive.

Let (Ω, F, P) be a probability space, where F is countably generated, and X be a Polish space. Let ϕ be a random dynamical system with time T on X. The skew product flow {Θ t , t ∈ T} induced by ϕ is a family of continuous operators acting on Pr Ω (X), the set of all probability measures on X ×Ω with marginal P, which is a Polish space equipped wit...

Let (Ω, F, P) be a probability space, where F is countably generated, and X be a Polish space. Let ϕ be a random dynamical system with time T on X. The skew product flow {Θ t , t ∈ T} induced by ϕ is a family of continuous operators acting on Pr Ω (X), the set of all probability measures on X ×Ω with marginal P, which is a Polish space equipped wit...

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 ∈ N such that T n (U) ∩ U = ∅. 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.

In this paper, we introduce and study the notions of hypercyclicity and transitivity for random dynamical systems and we establish the relation between them in a topological space. We also introduce the notions of mixing and weakly mixing for random dynamical systems and give some of their properties.

Let (F_Ω^n )n≥0 be the series of Faber polynomials associated with Ω, a bounded domain of the complex plane whose boundary is a closed Jordan curve. In this research, the Ω-hypercyclic and Ω-transitive behavior are studied within the framework of linear strongly continuous semigroups. We give sufficient constraints on the spectrum of an operator to...

An operator T acting on a separable complex Banach space $$\mathcal {B}$$ B is said to be hypercyclic if there exists $$f\in \mathcal {B}$$ f ∈ B such that the orbit $$\{T^n f:\ n\in \mathbb {N}\}$$ { T n f : n ∈ N } is dense in $$\mathcal {B}$$ B . Godefroy and Shapiro (J. Funct. Anal., 98(2):229–269, 1991) characterized those elements, which are...

A Banach space $X$ operator $T$ is called recurrent if the set $\{x\in X:\ x\in \overline{O(T,Tx)}\}$ is dense in $X$. The operator $T$ is weakly sequentially recurrent if the set $\{x\in X:\ x\in \overline{O(T,Tx)}^w\}$ is weakly dense in $X$. Costakis et al. \cite{reclinope} ask if $T\oplus T$ should be recurrent whenever $T$ is. This question ha...

A linear bounded operator T on a Hilbert space X is called hypercyclic if there exists a vector x ∈ X whose orbit under T ; {T n x; n ∈ N} is dense in X. The operator T is called recurrent (super-recurrent, respectively) if, for every non-empty open subset U ⊂ X, there is an integer n such that T n U ∩ U ̸ = / 0 (there is an integer n and a scalar...

A Furstenberg family F is a collection of infinite subsets of the set of positive integers such that if A⊂B and A∈F, then B∈F. For a Furstenberg family F, finitely many operators T1,...,TN acting on a common topological vector space X are said to be disjoint F-transitive if for every non-empty open subsets U0,...,UN of X the set {n∈N:U0∩T1-n(U1)∩.....

This paper is a continuation of our recent work on super-recurrence of operators [3]. We introduce and study subspace-super recurrence of operators. We give the relationship between this new class of operators, super recurrent operators, and other well known class of operators of linear topological dynamics. Several examples and proprieties are giv...

This paper studies the weakly sequentially recurrence property of shifts operators. In the case of $\ell^p(\mathbb{N})$, $1\leq p<\infty$, we show that the weak recurrence, recurrence, hypercyclicity, and weak hypercyclicity are equivalent. In the case of $\ell^\infty(\mathbb{N})$ (resp. $\ell^\infty(\mathbb{Z})$), we prove that the unilateral back...

In this paper, we generalize the notion of k-quasi-(m, n)-isosymmetric operators on Hilbert space defined by Sid Ahmed [21] when an additional semi-inner product is considered. We introduce the classe of p-quasi-(A, m, n)-isosymmetric operators. We study some basic properties of theses operators. We prove that theses operators have the SVEP propert...

We consider a Markov chain on a Polish space \(\textit{X}\) with Markov kernel \(\textit{P}\). This kernel is an operator acting on the space of all probability measures on \(\textit{X}\), which is a Polish space equipped with the narrow topology. In this work, we introduce and study the notion of the narrow recurrence of kernels of Markov chains a...

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...

A Banach space $X$ operator $T$ is called recurrent if the set $\{x\in X:\ x\in \overline{O(T,Tx)}\}$ is dense in $X$. The operator $T$ is weakly recurrent if the set $\{x\in X:\ x\in \overline{O(T,Tx)}^w\}$ is weakly dense in $X$. Costakis et al. \cite{reclinope} ask if $T\oplus T$ should be recurrent whenever $T$ is. This question has been answer...

This paper examines the phenomenon of super-recurrence exhibited by multiplication operators when applied to various function spaces. Our findings demonstrate that, contrary to hypercyclicity and supercyclicity cases, certain multiplication operators on specific function spaces exhibit super-recurrent behavior.

In this paper, we consider a generalization of $(A, m, n)$-isosymmetric operators on a Hilbert space to the multivariable setting. For a $d$-tuple of commuting operators $\mathbf{T}=(T_1, \dots, T_d)$, $A$ a positive operator, and $m, n$ are two positive integers, we introduce the class of $(A, m, n)$-isosymmetric $d$-tuple of commuting operators a...

In Morocco, 966,777 confirmed cases and 14,851 confirmed deaths because of COVID‐19 were recorded as of January 1, 2022. Recently, a new strain of COVID‐19, the so‐called Omicron variant, was reported in Morocco, which is considered to be more dangerous than the existing COVID‐19 virus. To end this ongoing global COVID‐19 pandemic and Omicron varia...

In this paper, we introduce and study the notion of weak recurrence on a Banach space X. In one hand, we prove some properties of this class of operators. On the other hand, we study the weak recurrence of power bounded and diagonal operators. This leads us to deduce the relationship between this notion and other notion in linear dynamics.

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 \(\lambda \in {\mathbb {K}}\) and \(n\in {\mathbb {N}}\) such that \(\lambda T^n(U)\cap U\ne \emptyset \). In this paper, we introduce and study the notions of super-rigidity and uniform super-rigidity which are related to the notion o...

In this paper, we investigate the notions of weak hypercyclicity and weak sequential hypercyclicity for strongly continuous semigroups of operators in Banach spaces.We study the relationship between these notions and hypercyclicity. We give sufficient conditions for a C 0-semigroup to be weakly hypercyclic and weakly sequentially hypercyclic. We pr...

In this paper, we introduce and study subspace weakly hypercyclic C 0-semigroups. We give sufficient conditions for a C 0-semigroup to be subspace weakly hypercyclic. We also give the relationship between this new notion of C 0-semigroups, weak hypercyclicity of C 0-semigroups, and hypercyclicity of C 0-semigroups. We characterize other properties...

In this paper we introduce and study the notion of subspace weak recurrence on a Banach space X. Firstly, we give the relationship between this notion on the others notions in linear dynamics. Secondly, we prove several properties of subspace weak recurrence operators. Some examples are also provided.

In this paper we introduce and study the notion of weak recurrence on a Banach space X. In one hand, we prove some properties of this class of operators. On the other hand, we study the weak recurrence of power bounded and diagonal operators. This leads us to deduce the relationship between this notion and other notion in linear dynamics.

In this paper, we study the weakly recurrence property of shifts operators. In the case of p (N), 2 ≤ p < ∞, we show that the weak recurrence, recurrence, hypercyclicity, and weak hypercyclicity are equivalent. In the case of ∞ (N) (resp. ∞ (Z)), we prove that the unilateral backward (resp. bilateral backward) can never be weakly recurrent.

Let X be a Banach space, B(X) the algebra of bounded linear operators on X and (J, · J) an admissible Banach ideal of B(X). For T ∈ B(X), let L J,T and R J,T ∈ B(J) denote the left and right multiplication defined by L J,T (A) = T A and R J,T (A) = AT , respectively. In this paper, we study the transmission of some concepts related to recurrent ope...

In this work, we examine super-recurrence and super-rigidity of composition operators acting on H(D) the space of holomorphic functions on the unit disk D and on H 2 (D) the Hardy-Hilbert space. We characterize the symbols that generate super-recurrent and super-rigid composition operators acting on H(D) and H 2 (D).

This paper investigates the novel coronavirus (COVID-19) infection system with a mathematical model presented with the Caputo derivative. We calculate equilibrium points and discuss their stability. We also go through the existence and uniqueness of a nonnegative solution for the system under study. We obtain numerical simulations for different ord...

Let (Ω, F , P) be a probability space and X be a Polish space equipped with its Borel σ-algebra B. We consider a transition function probability {P t , t ∈ R + } of a continuous Markov chain on (Ω, F , P) with values in X. This transition function defines a semi group acting on Pr(X), the set of all probability measures on X, which is also a Polish...

Let (Ω, F , P) be a probability space and X be a Polish space equipped with its Borel σ-algebra B. We consider a transition function probability {P t , t ∈ R + } of a continuous Markov chain on (Ω, F , P) with values in X. This transition function defines a semi group acting on Pr(X), the set of all probability measures on X, which is also a Polish...

A continuation of our recent work [6]. We introduce and and study in this paper subspace super-recurrent operators. We give the relationship between this new class of operators, super-recurrent operators, and other well known class of operators of linear topological dynamics. We establish a subspace super-recurrence criterion and we use to a give s...

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...

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 ∈ L(X) to a set of operators Γ ⊂ L(X). We prove some results for codiskcyclic sets of operators and we establish a codiskcyclicity criterion. As an application , we s...

In this paper, we study the notion of super-recurrence for a strongly continuous semigroup of operators. We establish some results for super-recurrent C0-semigroups. As an application, we study the super-recurrence of the translation C0-semigroup.

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 $\mathcal{F}$-transiti...

A bounded linear operator T acting on a Hilbert space H is said to be recurrent if for every non-empty open subset U⊂H there is an integer n such that Tn(U)∩U≠∅. 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...

In this work, we investigate super-recurrence, super-rigidity, and uniformly super-rigidity of composition operators acting on H(Ω) the space of holomorphic functions on Ω, where Ω is either the whole complex plane C or the punctured plane C\{0}. We deduce the form of the symbol φ that generates a super-recurrent, super-rigid, uniformly super-rigid...

Let X be a complex Banach space with dim X > 1 such that its topological dual X∗ is separable and B(X) the algebra of all bounded linear operators on X. In this paper, we study the passage of property of being supercyclic from T ∈ B(X) to the left and the right multiplication induced by T on an admissible Banach ideal of B(X). Also, we give a suffi...

This work is a continuation of our recent work on super-recurrence of strongly continuous semigroup [4]. We introduce and study subspace super-recurrent C 0-semigroups. We give sufficient conditions for a C 0-semigroup to be subspace super-recurrent. We also give the relationship between this new notion of C 0-semigroups, super-recurrence of C 0-se...

In this work, we investigate super-recurrence, super-rigidity, and uniformly super-rigidity of composition operators acting on H(Ω) the space of holomorphic functions on Ω, where Ω is either the complex plane C or the punctured plane C\{0}. We deduce the form of the symbol φ that generates a super-recurrent, super-rigid, uniformly super-rigid compo...

A bounded linear operator T acting on a Hilbert space H is said to be recurrent if for every non-empty open subset U ⊂ H there is an integer n such that T n (U) ∩ U = ∅. 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 D...

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 ∈ K and n ∈ N such that λT n (U) ∩ U = ∅. 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 cla...

A Furstenberg family \(\mathcal{F}\) is a collection of infinite subsets ofthe set of positive integers such that if \(A\subset B\) and \(A\in \mathcal{F}\), then \(B\in \mathcal{F}\). For aFurstenberg family \(\mathcal{F}\), an operator \(T\) on a topological vector space \(X\) is said tobe \(\mathcal{F}\)-transitive provided that for each non-emp...

In this paper, we propose a new epidemiological mathematical model for the spread of the COVID-19 disease with a special focus on the transmissibility of individuals with severe symptoms, mild symptoms, and asymptomatic symptoms. We compute the basic reproduction number and we study the local stability of the disease-free equilibrium in terms of th...

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 $\sigma(T)$ and $\sigma_p(T^*)$, the spectrum of $T$ and the p...

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 $\Gamma\subset L(X).$ We prove some results for codiskcyclic sets of operators and we establish a codiskcyclicity criterion. As...

Let X be a complex topological vector space and L(X) the set of all continuous linear operators on X. An operator T ? L(X) is supercyclic if there is x ? X such that, COrb(T,x) = {?Tnx : ? ? C, n ? 0}, is dense in X. In this paper, we extend this notion from a single operator T ? L(X) to a subset of operators ? ? L(X). We prove that most of related...

In this paper, we introduce and study the diskcyclicity and disk transitivity of a set of operators. We establish a diskcyclicity criterion and give the relationship between this criterion and the diskcyclicity. As applications, we study the diskcyclicty of C0-semigroups and C-regularized groups. We show that a diskcyclic C0-semigroup exists on a c...

A bounded linear operator T acting on a topological vector space X is called recurrent if for every nonempty open subset U ⊂ X there is an integer n such that T n U ∩U = ∅. In this work, we introduce the notion of quasi-periodicity in the context of topological vector space. This leads us to define a new class of operators in connection with recurr...

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 $\Gamma$ of operators. As application, we study the recurrence of $C$-regularized gro...

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 idea...

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...

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 kn...

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 kn...

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 operato...

Let $X$ be a complex topological vector space with $dim(X)>1$ and $\mathcal{B}(X)$ the space of all linear continuous operators on $X.$ In this paper, we extend the concept of cyclicity for a single operator and $C_0$-semigroups to a subset of $\mathcal{B}(X).$ We establish some results for cyclic set of operators and give some applications for $C_...

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.

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.

Let B(X) denote the algebra of all bounded linear operators on an infinitedimensional separable complex Banach space X and M be a nonzero subspace of X: We will characterize properties of being d-M mixing for a N ≥ 2 sequence T1;j ; T2;j. TN;j of operators in B(X): Also, we will give necessary and sufficient conditions for a N ≥ 2 sequence T1;j ; T...

In this paper, we continue the study of the pseudo B-Fredholm operators of Boasso, and the pseudo B-Weyl spectrum of Zariouh and Zguitti; in particular we find that the pseudo B-Weyl spectrum is empty whenever the pseudo B-Fredholm spectrum is, and look at the symmetric differences between the pseudo B-Weyl and other spectra.

Fuglede-Putnam Theorem has been proved for a considerably large number of class of operators. In this paper by using the spectral theory, we obtain a theoretical and general framework from which Fuglede-Putnam theorem may be promptly established for many classes of operators.

A Banach space operator satisfies generalized Rakočevi´Rakočevi´c's property (gw) if the complement of its upper semi B-Weyl spectrum in its approximate point spectrum is the set of eigenvalues of T which are isolated in the spectrum of T. In this note, we characterize hypecyclic and supercyclic operators satisfying the property (gw).

Let $(T(t))_{t\geq0}$ be a $C_0$ semigroups and $A$ be its infinitesimal generator. In this work, we prove that the spectral inclusion for $(T(t))_{t\geq0}$ remains true for the Drazin invertible and Quasi-Fredholm spectra. Also, we will give conditions under which facts $A$ is quasi-Fredholm, $A$ is Drazin invertible and $A$ is B-Fredholm are equi...

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.

Fuglede-Putnam Theorem have been proved for a considerably large number of class of operators. In this paper by using the spectral theory, we obtain a theoretical and general framework from which Fuglede-Putnam theorem may be promptly established for many classes of operators.

In this paper, we give conditions for which the $C_0$ semigroups satisfies spectral equality for semiregular, essentially semiregular and semi-Fredholm spectrum. Also, we establish the spectral inclusion for B-Fredholm spectrum of a $C_0$ semigroups.

In this paper, we give conditions for which the $C_0$ semigroups satisfies spectral equality for semiregular, essentially semiregular and semi-Fredholm spectrum. Also, we establish the spectral inclusion for B-Fredholm spectrum of a $C_0$ semigroups.

A Banach space operator satisfies property (gw) if the complement of its B-Weyl essential approximate point spectrum in its approximate point spectrum is the set of isolated eigenvalues of the operator. We give necessary and/or sufficient conditions ensuring the passage of property (gw) from two Banach space operators A and B to their tensor produc...

A Banach space operator T satises generalized a-Weyl's theorem if the complement of its upper semi B-Weyl spectrum in its approximate point spectrum is the set of eigenvalues of T which are isolated in the approximate spectrum of T: In this note we characterize hypecyclic and supercyclic operators satisfying generalized a-Weyl's theorem.

Let X and Y two complex Banach spaces and (A,B) a pair of bounded linear operators acting on X with value on Y. This paper is concerned with spectral analysis ofthe pair (A;B): We establish some properties concerning the spectrum of the linear operator pencils (A-lambda B) when B is not necessarily invertible and lambda is a complex number. Also, w...

Given Banach spaces $X$ and $Y$ and Banach space operators $A\in L(X)$ and $B\in L(Y)$, let $\rho\colon L(Y, X)\to L(Y, X)$ denote the generalized derivation defined by $A$ and $B$, i.e., $\rho (U)=AU-UB$ ($U\in L(Y, X)$). The main objective of this article is to study Weyl and Browder type theorems for $\rho\in L(L(Y, X))$. To this end, however, f...

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 \(\delta \in L(L(\mathcal {Y},\mathcal {X}))\) is defined by \(\delta (X)=AX-XB.\) In this article necessary and sufficient conditions ensuring that Rakočević’s property (w), on the o...

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 $\delta_{A,B} \in L(L(\mathcal{Y},\mathcal{X}))$ is defined by $\delta_{A,B}(X)=(L_{A}-R_{B})(X)=AX-XB$. This paper is concerned with the problem of the transferring the left polaroid property, f...

Given Banach spaces \({\mathcal{X}}\) and \({\mathcal{Y}}\) and Banach space operators \({A \in L (\mathcal{X})}\) and \({B \in L (\mathcal{Y})}\) , let \({\rho \colon L (\mathcal{Y},\mathcal{X})\to L(\mathcal{Y},\mathcal{X})}\) denote the generalized derivation defined by A and B, i.e., \({\rho (U)=AU-UB}\) (\({U\in L(\mathcal{Y},\mathcal{X})}\))....

Let L(X){\mathcal{L}(X)} be the algebra of all bounded linear operators on X and PS(X){\mathcal{P}S(X)} be the class of polaroid operators with the single-valued extension property. The property (gw) holds for T Î L(X){T \in \mathcal{L}(X)} if the complement in the approximate point spectrum of the semi-B-essential approximate point spectrum coinci...

Let $X$ a Banach space and $T$ a bounded linear operator on $X.$ We denote by $S(T)$ the set of all $\lambda \in \cit$ such that $T$ does not have the single-valued extension property at $\lambda$. In this note we prove equality up to $S(T)$ between the left Drazin spectrum and the left B-Fredholm spectrum and between the semi-essential approximate...

In this note, we characterize quasi-normality of two-sided multiplication, restricted to a norm ideal, and we extend this result to an important class which contains all quasi-normal operators. Also, we give some applications of this result.

Let T be a Banach space operator, E(T) be the set of all isolated eigenvalues of T and π(T) be the set of all poles of T. In this work, we show that Browder's theorem for T is equivalent to the localized single-valued extension property at all complex numbers λ in the complement of the Weyl spectrum of T, and we give some characterization of Weyl's...

In this note we introduce and study the property (gw), which extends property (w) introduced by Rakoc̆evic in [23]. We investigate the property (gw) in connection with Weyl type theorems. We show that if T is a bounded linear operator T acting on a Banach space X, then property (gw) holds for T if and only if property (w) holds for T and Π a (T) =...

In [4] we introduced and studied property (gw), which is an extension to the context of B-Fredholm theory, of the property (w) introduced by Rako˘ cevic in [28]. In this paper we continue the study of property (gw) and we consider its preservation under perturbations by finite rank and nilpotent operators.

Let T be a Banach space operator. In this paper, we characterize a-Browder’s theorem for T by the localized single valued extension property. Also, we characterize a-Weyl’s theorem under the condition E a (T)=π a (T), where E a (T) is the set of all eigenvalues of T which are isolated in the approximate point spectrum and π a (T) is the set of all...

Let A be a Banach algebra and L(A) the algebra of all bounded linear operators acting on A. For a; b 2 A, the generalized derivation ±a;b 2 L(A) and the elementary operator ¢a;b 2 L(A) are de¯ned by ±a;b(x) = ax ¡ xb and ¢a;b(x) = axb ¡ x, x 2 A. Let da;b = ±a;b or ¢a;b. In this note we give couples (a; b) 2 A2 such that the range and the kernel of...