Mohammed Benharrat | Ecole Nationale Polytechnique d'Oran (ENPO) (original) (raw)

Papers by Mohammed Benharrat

J. P. Labrousse [8] studied and characterized in the case of Hilbert spaces, a relation between t... more J. P. Labrousse [8] studied and characterized in the case of Hilbert spaces, a relation between the Kato essential spectrum (essential quasi-Fredholm spectrum) and another essential spectrum dened by ec(T) = { ∈ C;R(I − T) is not closed} (see [3]). In this paper, we investigate this relation in the case of Banach spaces.

Matematičnì studìï, Dec 27, 2019

We consider a linear operator pencil L(λ) = A − λB, λ ∈ C, where A and B are bounded operators on... more We consider a linear operator pencil L(λ) = A − λB, λ ∈ C, where A and B are bounded operators on Hilbert space. The purpose of this paper is to study the conditions under which the spectrum of L(.) is the whole complex plane or empty. This leads to some criteria for the spectrum to be bounded.

HAL (Le Centre pour la Communication Scientifique Directe), Jul 12, 2023

Let S be a n-by-n truncated shift whose numerical radius equal one. First, Cassier, Benharrat and... more Let S be a n-by-n truncated shift whose numerical radius equal one. First, Cassier, Benharrat and Belmouhoub in [12] proved that the Harnack part of S is trivial if n = 2, while, if n = 3, is an orbit associated with the action of a group of unitary diagonal matrices , see [12, Theorem 3.1 and Theorem 3.3]. Second, Cassier and Benharrat in [7] described elements of the Harnack part of the truncated n-by-n shift S under an extra assumption. In Section 2, we present useful results in the general finite dimensional situation. In Section 3, we give a complete description of the Harnack part of S for n = 4, the answer is surprising and instructive. It shows that, even when the dimension is an odd number, the Harnack part is bigger than conjectured in [7, Question 2.]. We also give a negative answer to [7, Question 1.] when ρ = 2.

arXiv (Cornell University), Sep 19, 2021

We establish some relationships between an m-accretive operator and its Moore-Penorse inverse. We... more We establish some relationships between an m-accretive operator and its Moore-Penorse inverse. We derive some perturbation result of the Moore-Penorse inverse of a maximal accretive operator. As an application we give a factorization theorem for a quadratic pencil of accretive operators. Also, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation. Illustrative examples are also given.

arXiv (Cornell University), May 3, 2023

In this paper we explore the theory of fractional powers of maximal accretive operators to obtain... more In this paper we explore the theory of fractional powers of maximal accretive operators to obtain results of existence, regularity and behavior asymptotic of solutions for linear abstract evolution equations of third order in time.

Eurasian Mathematical Journal

FsF furenkovD wF ytelevD FeF dovnihy ViceEditorsinChief uFxF yspnovD FF rrykov Editors hFeF elimo... more FsF furenkovD wF ytelevD FeF dovnihy ViceEditorsinChief uFxF yspnovD FF rrykov Editors hFeF elimov @zekistnAD rF fegehr @qermnyAD F fekjn @ghinAD yFF fesov @ussiAD xFuF fliev @uzkhstnAD xFeF fokyev @uzkhstnAD eFeF foruev @uyrgyzstnAD qF fourdud @prneAD eF getno @ortuglAD wF grro @pinAD eFhFF ghoudry @kistnAD FxF ghurikov @ussiAD eFF hzumdildev @uzkhstnAD FwF pilippov @ussiAD rF qhzryn @ermeniAD wFvF qoldmn @ussiAD F qoldshtein @ssrelAD F quliyev @ezerijnAD hFhF rroske @qermnyAD eF rsnoglu @urkeyAD wF ruxley @qret fritinAD F tin @sndiAD FhF ulmenov @uzkhstnAD fFiF ungyzhin @uzkhstnAD uFuF uenzhiev @uzkhstnAD FxF uhrin @uzkhstnAD iF uissin @qret fritinAD F uokilshvili @qeorgiAD FsF uorzyuk @felrusAD eF uufner @gzeh epuliAD vFuF uussinov @uzkhstnAD FhF vmerti @stlyAD wF vnz de gristoforis @stlyAD pF vnE zr @stlyAD FqF wz9y @wedenAD uFF wynyev @uzkhstnAD iFhF xursultnov @uzkhstnAD F yinrov @uzkhstnAD sFxF rsidis @qreeeAD tF e § ri¡ @grotiAD FeF lks @krineAD vFE iF ersson @wedenAD iFvF resmn @ussiAD wFeF gus @stlyAD wFhF mznov @ussiAD wF eissig @qermnyAD wF uzhnsky @qret fritinAD wFeF dyekov @uzkhstnAD F gitov @wedenAD FyF hposhnikov @wedenAD eFeF hklikov @ussiAD FeF kvortsov @olndAD qF inE nmon @gndAD iFF milov @uzkhstnAD FhF tepnov @ussiAD FF ultnev @ussiAD hF urgn @uzkhstnAD sFeF imnov @ussiAD tFeF ussupov @uzkhstnAD FF mirev @uzkhstnAD FhF smnov @jikistnAD xF silevski @wexioAD hhun ng @ghinAD fFF huE mgulov @uzkhstnA Managing Editor eFwF emirkhnov c he vFxF qumilyov iursin xtionl niversity

We consider an abstract optimal control problem with additional equality and inequality state and... more We consider an abstract optimal control problem with additional equality and inequality state and control constraints, we use the exact penalty function to transform the constrained optimal control problem into an unconstrained one. We establishes, under suitable assumptions, an equivalence between the two problems in the “local” sense. Sufficient conditions for a penalty function to be exact are given.

Proyecciones, Dec 1, 2019

In this paper, we give some characterizations of the left and right generalized Drazin invertible... more In this paper, we give some characterizations of the left and right generalized Drazin invertible bounded operators in Banach spaces by means of the single-valued extension property (SVEP). In particular, we show that a bounded operator is left (resp. right) generalized Drazin invertible if and only if admits a generalized Kato decomposition and has the SVEP at 0 (resp. it admits a generalized Kato decomposition and its adjoint has the SVEP at 0. In addition, we prove that both of the left and the right generalized Drazin operators are invariant under additive commuting finite rank perturbations. Furthermore, we investigate the transmission of some local spectral properties from a bounded linear operator, as the SVEP, Dunford property (C), and property (β), to its generalized Drazin inverse. 2010 Mathematics Subject Classification. 47A10. Key words and phrases. Left and right generalized Drazin invertible operators; Generalized Drazin invertible operators; Generalized Kato decomposition; SVEP; Local spectral theory.

arXiv (Cornell University), Dec 6, 2019

The purpose of this paper is to analysis the Harnack part of some truncated shifts whose ρ-numeri... more The purpose of this paper is to analysis the Harnack part of some truncated shifts whose ρ-numerical radius equal one in the finite dimensional case. As pointed out in Theorem 1.17 [12], a key point is to describe the null spaces of the ρ-operatorial kernel of these truncated shifts. We establish two fundamental results in this direction and some applications are also given.

Tamkang Journal of Mathematics

The main purpose of this paper is to give an improvement of numerical radius inequality for an up... more The main purpose of this paper is to give an improvement of numerical radius inequality for an upper triangular operator matrix.

International Journal of Analysis and Applications, 2015

This paper is devoted to the investigation of the stability of the Weyl essential spectrum of clo... more This paper is devoted to the investigation of the stability of the Weyl essential spectrum of closed densely dened linear operator A subjected to additive perturbation K such that (lambda-A-K)^{-1}K or K(lambda-A-K)^{-1} is a quasi-compact operator. The obtained results are used to describe the Weyl essential spectrum of singular neutron transport operator.

Filomat

We establish some relationships between an m-accretive operator and its Moore-Penorse inverse. We... more We establish some relationships between an m-accretive operator and its Moore-Penorse inverse. We derive some perturbation result the Moore-Penorse inverse of a maximal accretive operator. As an application we give a factorization theorem for a quadratic pencil of accretive operators. Also, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation. Illustrative examples are also given.

arXiv (Cornell University), Dec 17, 2016

The purpose of this paper is to describe the Harnack parts for the operators of class C ρ (ρ > 0)... more The purpose of this paper is to describe the Harnack parts for the operators of class C ρ (ρ > 0) on Hilbert spaces which were introduced by B. Sz. Nagy and C. Foias in [25]. More precisely, we study Harnack parts of operators with ρ-numerical radius one. The case of operators with ρ-numerical radius strictly less than 1 was described in [10]. We obtain a general criterion for compact ρ-contractions to be in the same Harnack part. We give a useful equivalent form of this criterion for usual contractions. Operators with numerical radius one received also a particular attention. Moreover, we study many properties of Harnack equivalence in the general case.

arXiv (Cornell University), Aug 10, 2021

We prove, in some cases in term of kippenhahn curve, that if 5-by-5 partial isometry whose numeri... more We prove, in some cases in term of kippenhahn curve, that if 5-by-5 partial isometry whose numerical range is a circular disc then its center is must be the origin. This gives a partial affirmative answer of the Conjecture 5.1. of [H. l. Gau et al., Linear and Multilinear Algebra, 64 (1) 2016, 14-35.], for the five dimensional case.

We prove necessary optimality conditions of Euler-Lagrange type for a problem of the calculus of ... more We prove necessary optimality conditions of Euler-Lagrange type for a problem of the calculus of variations with time delays, where the delay in the unknown function is different from the delay in its derivative. Then, a more general optimal control problem with time delays is considered. Main result gives a convergence theorem, allowing to obtain a solution to the delayed optimal control problem by considering a sequence of delayed problems of the calculus of variations.

arXiv: Functional Analysis, 2016

The purpose of this paper is to describe the Harnack parts for the operators of class C rho\rhorho ($... more The purpose of this paper is to describe the Harnack parts for the operators of class C rho\rhorho ($\rho$ \textgreater{} 0) on Hilbert spaces which were introduced by B. Sz. Nagy and C. Foias in [25]. More precisely, we study Harnack parts of operators with rho\rhorho-numerical radius one. The case of operators with rho\rhorho-numerical radius strictly less than 1 was described in [10]. We obtain a general criterion for compact rho\rhorho-contractions to be in the same Harnack part. We give a useful equivalent form of this criterion for usual contractions. Operators with numerical radius one received also a particular attention. Moreover, we study many properties of Harnack equivalence in the general case.

International Journal of Analysis and Applications, 2015

To be able to refine the completion of C(H1, H2), the of set all closed densely defined linear op... more To be able to refine the completion of C(H1, H2), the of set all closed densely defined linear operators between two Hilbert spaces H1 and H2, we define in this paper some new strictly stronger metrics than the gap metric g and we characterize the closure with respect to theses metrics of the subset L(H1, H2) of bounded elements of C(H1, H2). In addition, several operator norm inequalities concerning the equivalence of some metrics on L(H1, H2) are presented. We also establish the semi-Fredholmness and Fredholmness of unbounded in terms of bounded pure contractions.

In this paper we attempt to investigate some algebraic and topological properties of quotient ope... more In this paper we attempt to investigate some algebraic and topological properties of quotient operators acting on Hilbert space, and to give a characterization of Fredholm quotient operator and its index.

arXiv: Functional Analysis, 2020

A new sufficient condition is given for the sum of linear m-accretive operator and accretive oper... more A new sufficient condition is given for the sum of linear m-accretive operator and accretive operator one in a Hilbert space to be m-accretive. As an application, an extended result to the operator-norm error bound estimate for the exponential Trotter-Kato product formula is given.

J. P. Labrousse [8] studied and characterized in the case of Hilbert spaces, a relation between t... more J. P. Labrousse [8] studied and characterized in the case of Hilbert spaces, a relation between the Kato essential spectrum (essential quasi-Fredholm spectrum) and another essential spectrum dened by ec(T) = { ∈ C;R(I − T) is not closed} (see [3]). In this paper, we investigate this relation in the case of Banach spaces.

Matematičnì studìï, Dec 27, 2019

We consider a linear operator pencil L(λ) = A − λB, λ ∈ C, where A and B are bounded operators on... more We consider a linear operator pencil L(λ) = A − λB, λ ∈ C, where A and B are bounded operators on Hilbert space. The purpose of this paper is to study the conditions under which the spectrum of L(.) is the whole complex plane or empty. This leads to some criteria for the spectrum to be bounded.

HAL (Le Centre pour la Communication Scientifique Directe), Jul 12, 2023

Let S be a n-by-n truncated shift whose numerical radius equal one. First, Cassier, Benharrat and... more Let S be a n-by-n truncated shift whose numerical radius equal one. First, Cassier, Benharrat and Belmouhoub in [12] proved that the Harnack part of S is trivial if n = 2, while, if n = 3, is an orbit associated with the action of a group of unitary diagonal matrices , see [12, Theorem 3.1 and Theorem 3.3]. Second, Cassier and Benharrat in [7] described elements of the Harnack part of the truncated n-by-n shift S under an extra assumption. In Section 2, we present useful results in the general finite dimensional situation. In Section 3, we give a complete description of the Harnack part of S for n = 4, the answer is surprising and instructive. It shows that, even when the dimension is an odd number, the Harnack part is bigger than conjectured in [7, Question 2.]. We also give a negative answer to [7, Question 1.] when ρ = 2.

arXiv (Cornell University), Sep 19, 2021

We establish some relationships between an m-accretive operator and its Moore-Penorse inverse. We... more We establish some relationships between an m-accretive operator and its Moore-Penorse inverse. We derive some perturbation result of the Moore-Penorse inverse of a maximal accretive operator. As an application we give a factorization theorem for a quadratic pencil of accretive operators. Also, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation. Illustrative examples are also given.

arXiv (Cornell University), May 3, 2023

In this paper we explore the theory of fractional powers of maximal accretive operators to obtain... more In this paper we explore the theory of fractional powers of maximal accretive operators to obtain results of existence, regularity and behavior asymptotic of solutions for linear abstract evolution equations of third order in time.

Eurasian Mathematical Journal

FsF furenkovD wF ytelevD FeF dovnihy ViceEditorsinChief uFxF yspnovD FF rrykov Editors hFeF elimo... more FsF furenkovD wF ytelevD FeF dovnihy ViceEditorsinChief uFxF yspnovD FF rrykov Editors hFeF elimov @zekistnAD rF fegehr @qermnyAD F fekjn @ghinAD yFF fesov @ussiAD xFuF fliev @uzkhstnAD xFeF fokyev @uzkhstnAD eFeF foruev @uyrgyzstnAD qF fourdud @prneAD eF getno @ortuglAD wF grro @pinAD eFhFF ghoudry @kistnAD FxF ghurikov @ussiAD eFF hzumdildev @uzkhstnAD FwF pilippov @ussiAD rF qhzryn @ermeniAD wFvF qoldmn @ussiAD F qoldshtein @ssrelAD F quliyev @ezerijnAD hFhF rroske @qermnyAD eF rsnoglu @urkeyAD wF ruxley @qret fritinAD F tin @sndiAD FhF ulmenov @uzkhstnAD fFiF ungyzhin @uzkhstnAD uFuF uenzhiev @uzkhstnAD FxF uhrin @uzkhstnAD iF uissin @qret fritinAD F uokilshvili @qeorgiAD FsF uorzyuk @felrusAD eF uufner @gzeh epuliAD vFuF uussinov @uzkhstnAD FhF vmerti @stlyAD wF vnz de gristoforis @stlyAD pF vnE zr @stlyAD FqF wz9y @wedenAD uFF wynyev @uzkhstnAD iFhF xursultnov @uzkhstnAD F yinrov @uzkhstnAD sFxF rsidis @qreeeAD tF e § ri¡ @grotiAD FeF lks @krineAD vFE iF ersson @wedenAD iFvF resmn @ussiAD wFeF gus @stlyAD wFhF mznov @ussiAD wF eissig @qermnyAD wF uzhnsky @qret fritinAD wFeF dyekov @uzkhstnAD F gitov @wedenAD FyF hposhnikov @wedenAD eFeF hklikov @ussiAD FeF kvortsov @olndAD qF inE nmon @gndAD iFF milov @uzkhstnAD FhF tepnov @ussiAD FF ultnev @ussiAD hF urgn @uzkhstnAD sFeF imnov @ussiAD tFeF ussupov @uzkhstnAD FF mirev @uzkhstnAD FhF smnov @jikistnAD xF silevski @wexioAD hhun ng @ghinAD fFF huE mgulov @uzkhstnA Managing Editor eFwF emirkhnov c he vFxF qumilyov iursin xtionl niversity

We consider an abstract optimal control problem with additional equality and inequality state and... more We consider an abstract optimal control problem with additional equality and inequality state and control constraints, we use the exact penalty function to transform the constrained optimal control problem into an unconstrained one. We establishes, under suitable assumptions, an equivalence between the two problems in the “local” sense. Sufficient conditions for a penalty function to be exact are given.

Proyecciones, Dec 1, 2019

In this paper, we give some characterizations of the left and right generalized Drazin invertible... more In this paper, we give some characterizations of the left and right generalized Drazin invertible bounded operators in Banach spaces by means of the single-valued extension property (SVEP). In particular, we show that a bounded operator is left (resp. right) generalized Drazin invertible if and only if admits a generalized Kato decomposition and has the SVEP at 0 (resp. it admits a generalized Kato decomposition and its adjoint has the SVEP at 0. In addition, we prove that both of the left and the right generalized Drazin operators are invariant under additive commuting finite rank perturbations. Furthermore, we investigate the transmission of some local spectral properties from a bounded linear operator, as the SVEP, Dunford property (C), and property (β), to its generalized Drazin inverse. 2010 Mathematics Subject Classification. 47A10. Key words and phrases. Left and right generalized Drazin invertible operators; Generalized Drazin invertible operators; Generalized Kato decomposition; SVEP; Local spectral theory.

arXiv (Cornell University), Dec 6, 2019

The purpose of this paper is to analysis the Harnack part of some truncated shifts whose ρ-numeri... more The purpose of this paper is to analysis the Harnack part of some truncated shifts whose ρ-numerical radius equal one in the finite dimensional case. As pointed out in Theorem 1.17 [12], a key point is to describe the null spaces of the ρ-operatorial kernel of these truncated shifts. We establish two fundamental results in this direction and some applications are also given.

Tamkang Journal of Mathematics

The main purpose of this paper is to give an improvement of numerical radius inequality for an up... more The main purpose of this paper is to give an improvement of numerical radius inequality for an upper triangular operator matrix.

International Journal of Analysis and Applications, 2015

This paper is devoted to the investigation of the stability of the Weyl essential spectrum of clo... more This paper is devoted to the investigation of the stability of the Weyl essential spectrum of closed densely dened linear operator A subjected to additive perturbation K such that (lambda-A-K)^{-1}K or K(lambda-A-K)^{-1} is a quasi-compact operator. The obtained results are used to describe the Weyl essential spectrum of singular neutron transport operator.

Filomat

We establish some relationships between an m-accretive operator and its Moore-Penorse inverse. We... more We establish some relationships between an m-accretive operator and its Moore-Penorse inverse. We derive some perturbation result the Moore-Penorse inverse of a maximal accretive operator. As an application we give a factorization theorem for a quadratic pencil of accretive operators. Also, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation. Illustrative examples are also given.

arXiv (Cornell University), Dec 17, 2016

The purpose of this paper is to describe the Harnack parts for the operators of class C ρ (ρ > 0)... more The purpose of this paper is to describe the Harnack parts for the operators of class C ρ (ρ > 0) on Hilbert spaces which were introduced by B. Sz. Nagy and C. Foias in [25]. More precisely, we study Harnack parts of operators with ρ-numerical radius one. The case of operators with ρ-numerical radius strictly less than 1 was described in [10]. We obtain a general criterion for compact ρ-contractions to be in the same Harnack part. We give a useful equivalent form of this criterion for usual contractions. Operators with numerical radius one received also a particular attention. Moreover, we study many properties of Harnack equivalence in the general case.

arXiv (Cornell University), Aug 10, 2021

We prove, in some cases in term of kippenhahn curve, that if 5-by-5 partial isometry whose numeri... more We prove, in some cases in term of kippenhahn curve, that if 5-by-5 partial isometry whose numerical range is a circular disc then its center is must be the origin. This gives a partial affirmative answer of the Conjecture 5.1. of [H. l. Gau et al., Linear and Multilinear Algebra, 64 (1) 2016, 14-35.], for the five dimensional case.

We prove necessary optimality conditions of Euler-Lagrange type for a problem of the calculus of ... more We prove necessary optimality conditions of Euler-Lagrange type for a problem of the calculus of variations with time delays, where the delay in the unknown function is different from the delay in its derivative. Then, a more general optimal control problem with time delays is considered. Main result gives a convergence theorem, allowing to obtain a solution to the delayed optimal control problem by considering a sequence of delayed problems of the calculus of variations.

arXiv: Functional Analysis, 2016

The purpose of this paper is to describe the Harnack parts for the operators of class C rho\rhorho ($... more The purpose of this paper is to describe the Harnack parts for the operators of class C rho\rhorho ($\rho$ \textgreater{} 0) on Hilbert spaces which were introduced by B. Sz. Nagy and C. Foias in [25]. More precisely, we study Harnack parts of operators with rho\rhorho-numerical radius one. The case of operators with rho\rhorho-numerical radius strictly less than 1 was described in [10]. We obtain a general criterion for compact rho\rhorho-contractions to be in the same Harnack part. We give a useful equivalent form of this criterion for usual contractions. Operators with numerical radius one received also a particular attention. Moreover, we study many properties of Harnack equivalence in the general case.

International Journal of Analysis and Applications, 2015

To be able to refine the completion of C(H1, H2), the of set all closed densely defined linear op... more To be able to refine the completion of C(H1, H2), the of set all closed densely defined linear operators between two Hilbert spaces H1 and H2, we define in this paper some new strictly stronger metrics than the gap metric g and we characterize the closure with respect to theses metrics of the subset L(H1, H2) of bounded elements of C(H1, H2). In addition, several operator norm inequalities concerning the equivalence of some metrics on L(H1, H2) are presented. We also establish the semi-Fredholmness and Fredholmness of unbounded in terms of bounded pure contractions.

In this paper we attempt to investigate some algebraic and topological properties of quotient ope... more In this paper we attempt to investigate some algebraic and topological properties of quotient operators acting on Hilbert space, and to give a characterization of Fredholm quotient operator and its index.

arXiv: Functional Analysis, 2020

A new sufficient condition is given for the sum of linear m-accretive operator and accretive oper... more A new sufficient condition is given for the sum of linear m-accretive operator and accretive operator one in a Hilbert space to be m-accretive. As an application, an extended result to the operator-norm error bound estimate for the exponential Trotter-Kato product formula is given.