Ameur Seddik - Academia.edu (original) (raw)

Papers by Ameur Seddik

Mathematical inequalities & applications, 2024

Advances in Operator Theory, 2023

In this survey, we shall present the characterizations of some distinguished classes of bounded ... more In this survey, we shall present the characterizations of some distinguished classes
of bounded linear operators acting on a complex separable Hilbert space in terms of
operator inequalities related to the arithmetic–geometric mean inequality.

Acta Scientiarum Mathematicarum, 2018

In this note, we present several characterizations for some distinguished classes of bounded Hilb... more In this note, we present several characterizations for some distinguished classes of bounded Hilbert space operators (self-adjoint operators, normal operators, unitary operators, and isometry operators) in terms of operator inequalities.

In this note, we shall give some characterizations of some distinguished classes of operators (se... more In this note, we shall give some characterizations of some distinguished classes of operators (selfadjoint, normal, unitary, partial isometry and isometry) in terms of operator inequalities.

arXiv: Functional Analysis, 2020

In this survey, we shall present characterizations of some distinguished classes of bounded linea... more In this survey, we shall present characterizations of some distinguished classes of bounded linear operators acting on a complex Hilbert space in terms of operator inequalities related to the arithmetic-geometric mean inequality.

In this note, we shall give complete characterizations of the class of all normal operators with ... more In this note, we shall give complete characterizations of the class of all normal operators with closed range, and the class of all selfadjoint operators with closed range multiplied by scalars in terms of some operator inequalities.

Let B(H) and A be a C∗−algebra of all bounded linear operators on a complex Hilbert space H and a... more Let B(H) and A be a C∗−algebra of all bounded linear operators on a complex Hilbert space H and a complex normed algebra, respectively. For A,B ∈ A, define a basic elementary operator MA,B : A→ A by MA,B(X) = AXB. An elementary operator is a finite sum RA,B = n P i=1 MAi,Bi of the basic ones, where A = (A1, ..., An) and B = (B1, ..., Bn) are two n-tuples of elements of A. If A is a standard operator algebra of B(H), it is proved that: (i) [4] °°MA,B + MB,A°° ≥ 2(√2− 1) kAk kBk , for any A,B ∈ A (ii)[1 ] °°MA,B + MB,A°° ≥ kAk kBk , for A,B ∈ A, such that inf λ∈C kA + λBk = kAk or inf λ∈C kB + λAk = kBk , (iii)[3] °°MA,B + MB,A°° = 2 kAk kBk , if kA + λBk = kAk+kBk , for some unit scalar λ. In this note, we are interested in the general situation where A is a standard operator algebra acting on a normed space. We shall prove that °°RA,B°° ≥ sup f,g∈(A∗)1 ̄̄̄̄ n P i=1 f(Ai)g(Bi) ̄̄̄̄ , for any two n-tuples A = (A1, ..., An) and B = (B1, ...,Bn) of elements of A (where (A∗)1 is the unit...

We correct a mistake which affect our main results, namely the proof of Lema 1. The main results ... more We correct a mistake which affect our main results, namely the proof of Lema 1. The main results of the article remain unchanged

arXiv: Functional Analysis, 2020

In this survey, we shall present characterizations of some distinguished classes of Hilbertian bo... more In this survey, we shall present characterizations of some distinguished classes of Hilbertian bounded linear operators (namely, normal operators, selfadjoint operators, and unitary operators) in terms of operator inequalities related to the arithmetic-geometric mean inequality. For the class of all normal operators, we shall present new general characterizations.

Operators and Matrices

Let B(H) be the C *-algebra of all bounded linear operators acting on a complex separable Hilbert... more Let B(H) be the C *-algebra of all bounded linear operators acting on a complex separable Hilbert space H. We shall show that: 1. The class of all selfadjoint operators in B(H) multiplied by scalars is characterized by ∀X ∈ B(H), S 2 X + XS 2 2 SXS , (S ∈ B(H)). 2. The class of all normal operators in B(H) is characterized by each of the three following properties (where D S = S * S − SS * , for S ∈ B(H)), (i) ∀X ∈ B(H), S 2 X + XS 2 2 SXS ,(S ∈ B(H)) , (

Linear Algebra and its Applications

Banach Journal of Mathematical Analysis, 2012

Linear Algebra Appl, 1998

Linear Algebra and its Applications

Let H 1 , H 2 be Hilbert spaces, A∈L(H 1 ), B∈L(H 2 ), and δ A,B (x)=AX-XB,∀X∈L(H 2 ,H 1 )· Among... more Let H 1 , H 2 be Hilbert spaces, A∈L(H 1 ), B∈L(H 2 ), and δ A,B (x)=AX-XB,∀X∈L(H 2 ,H 1 )· Among other results, the authors prove that: If there exists a quadratic polynomial p such that p(A) and p(B) are normal, then R(δ A,B ) ¯∩Ker(δ A * ,B * )={0}; where R(δ A,B ) ¯ is the closure of the image R(δ A,B ) of δ A,B . When A=B, we get a result of Yang Ho.

Proceedings of the American Mathematical Society

The problem considered here is that of finding (i) some consequences of the Corach-Porta-Recht In... more The problem considered here is that of finding (i) some consequences of the Corach-Porta-Recht Inequality; (ii) a necessary condition (resp. necessary and sufficient condition, when σ(P) = σ(Q)) for the invertible positive operators P, Q to satisfy the operator-norm inequality P XP −1 + Q −1 XQ ≥ 2 X , for all X in L(H); (iii) a necessary and sufficient condition for the invertible operator S in L(H) to satisfy (*) .

Given a standard operator algebra A acting on a complex normed space and A, B ∈ A we have: (i) Th... more Given a standard operator algebra A acting on a complex normed space and A, B ∈ A we have: (i) The lower estimate°°M A,B + M B,A°°≥ 2(√ 2 − 1) kAk kBk holds. (ii) The lower estimate°°M A,B + M B,A°°≥ kAk kBk holds if inf λ∈C kA + λBk = kAk or inf λ∈C kB + λAk = kBk. (iii) The equality°°M A,B + M B,A°°= 2 kAk kBk holds if kA + λBk = kAk + kBk for some unit scalar λ. These results extend analogous estimates establishes earier for standard operator subalgebras of Hilbert space operators.

Integral Equations and Operator Theory

VX e s fA(X) = AX - XA; we denote R(~A), R(~A)- and {A}' respectively the range, the norm clo... more VX e s fA(X) = AX - XA; we denote R(~A), R(~A)- and {A}' respectively the range, the norm closure of the range and the kernel of ~A. We denote Af = {A e/:(g) : R(~A)- (2 {A*}' = {0}}. If H is finite-dimensional, Af = s If H is infinite-dimensional, this equality does not hold. So a reasonable purpose is to determine what elements are in N. When H is a separable Hilbert space, AY contains the operators A for which p(A) is normal for some quadratic polynomial p(z) [2 ],the subnormal operators with cyclic vectors [2 ] and the ison:letries [3 ]. In this paper, we show that AY contains also all the operators unitarity equivalent to Jordan operators.

Banach Journal of Mathematical Analysis, 2012

In the present paper, taking some advantages offered by the context of finite dimensional Hilbert... more In the present paper, taking some advantages offered by the context of finite dimensional Hilbert spaces, we shall give a complete characterizations of certain distinguished classes of operators (self-adjoint, unitary reflection, normal) in terms of operator inequalities. These results extend previous characterizations obtained by the second author.

Mathematical inequalities & applications, 2024

Advances in Operator Theory, 2023

In this survey, we shall present the characterizations of some distinguished classes of bounded ... more In this survey, we shall present the characterizations of some distinguished classes
of bounded linear operators acting on a complex separable Hilbert space in terms of
operator inequalities related to the arithmetic–geometric mean inequality.

Acta Scientiarum Mathematicarum, 2018

In this note, we present several characterizations for some distinguished classes of bounded Hilb... more In this note, we present several characterizations for some distinguished classes of bounded Hilbert space operators (self-adjoint operators, normal operators, unitary operators, and isometry operators) in terms of operator inequalities.

In this note, we shall give some characterizations of some distinguished classes of operators (se... more In this note, we shall give some characterizations of some distinguished classes of operators (selfadjoint, normal, unitary, partial isometry and isometry) in terms of operator inequalities.

arXiv: Functional Analysis, 2020

In this survey, we shall present characterizations of some distinguished classes of bounded linea... more In this survey, we shall present characterizations of some distinguished classes of bounded linear operators acting on a complex Hilbert space in terms of operator inequalities related to the arithmetic-geometric mean inequality.

In this note, we shall give complete characterizations of the class of all normal operators with ... more In this note, we shall give complete characterizations of the class of all normal operators with closed range, and the class of all selfadjoint operators with closed range multiplied by scalars in terms of some operator inequalities.

Let B(H) and A be a C∗−algebra of all bounded linear operators on a complex Hilbert space H and a... more Let B(H) and A be a C∗−algebra of all bounded linear operators on a complex Hilbert space H and a complex normed algebra, respectively. For A,B ∈ A, define a basic elementary operator MA,B : A→ A by MA,B(X) = AXB. An elementary operator is a finite sum RA,B = n P i=1 MAi,Bi of the basic ones, where A = (A1, ..., An) and B = (B1, ..., Bn) are two n-tuples of elements of A. If A is a standard operator algebra of B(H), it is proved that: (i) [4] °°MA,B + MB,A°° ≥ 2(√2− 1) kAk kBk , for any A,B ∈ A (ii)[1 ] °°MA,B + MB,A°° ≥ kAk kBk , for A,B ∈ A, such that inf λ∈C kA + λBk = kAk or inf λ∈C kB + λAk = kBk , (iii)[3] °°MA,B + MB,A°° = 2 kAk kBk , if kA + λBk = kAk+kBk , for some unit scalar λ. In this note, we are interested in the general situation where A is a standard operator algebra acting on a normed space. We shall prove that °°RA,B°° ≥ sup f,g∈(A∗)1 ̄̄̄̄ n P i=1 f(Ai)g(Bi) ̄̄̄̄ , for any two n-tuples A = (A1, ..., An) and B = (B1, ...,Bn) of elements of A (where (A∗)1 is the unit...

We correct a mistake which affect our main results, namely the proof of Lema 1. The main results ... more We correct a mistake which affect our main results, namely the proof of Lema 1. The main results of the article remain unchanged

arXiv: Functional Analysis, 2020

In this survey, we shall present characterizations of some distinguished classes of Hilbertian bo... more In this survey, we shall present characterizations of some distinguished classes of Hilbertian bounded linear operators (namely, normal operators, selfadjoint operators, and unitary operators) in terms of operator inequalities related to the arithmetic-geometric mean inequality. For the class of all normal operators, we shall present new general characterizations.

Operators and Matrices

Let B(H) be the C *-algebra of all bounded linear operators acting on a complex separable Hilbert... more Let B(H) be the C *-algebra of all bounded linear operators acting on a complex separable Hilbert space H. We shall show that: 1. The class of all selfadjoint operators in B(H) multiplied by scalars is characterized by ∀X ∈ B(H), S 2 X + XS 2 2 SXS , (S ∈ B(H)). 2. The class of all normal operators in B(H) is characterized by each of the three following properties (where D S = S * S − SS * , for S ∈ B(H)), (i) ∀X ∈ B(H), S 2 X + XS 2 2 SXS ,(S ∈ B(H)) , (

Linear Algebra and its Applications

Banach Journal of Mathematical Analysis, 2012

Linear Algebra Appl, 1998

Linear Algebra and its Applications

Let H 1 , H 2 be Hilbert spaces, A∈L(H 1 ), B∈L(H 2 ), and δ A,B (x)=AX-XB,∀X∈L(H 2 ,H 1 )· Among... more Let H 1 , H 2 be Hilbert spaces, A∈L(H 1 ), B∈L(H 2 ), and δ A,B (x)=AX-XB,∀X∈L(H 2 ,H 1 )· Among other results, the authors prove that: If there exists a quadratic polynomial p such that p(A) and p(B) are normal, then R(δ A,B ) ¯∩Ker(δ A * ,B * )={0}; where R(δ A,B ) ¯ is the closure of the image R(δ A,B ) of δ A,B . When A=B, we get a result of Yang Ho.

Proceedings of the American Mathematical Society

The problem considered here is that of finding (i) some consequences of the Corach-Porta-Recht In... more The problem considered here is that of finding (i) some consequences of the Corach-Porta-Recht Inequality; (ii) a necessary condition (resp. necessary and sufficient condition, when σ(P) = σ(Q)) for the invertible positive operators P, Q to satisfy the operator-norm inequality P XP −1 + Q −1 XQ ≥ 2 X , for all X in L(H); (iii) a necessary and sufficient condition for the invertible operator S in L(H) to satisfy (*) .

Given a standard operator algebra A acting on a complex normed space and A, B ∈ A we have: (i) Th... more Given a standard operator algebra A acting on a complex normed space and A, B ∈ A we have: (i) The lower estimate°°M A,B + M B,A°°≥ 2(√ 2 − 1) kAk kBk holds. (ii) The lower estimate°°M A,B + M B,A°°≥ kAk kBk holds if inf λ∈C kA + λBk = kAk or inf λ∈C kB + λAk = kBk. (iii) The equality°°M A,B + M B,A°°= 2 kAk kBk holds if kA + λBk = kAk + kBk for some unit scalar λ. These results extend analogous estimates establishes earier for standard operator subalgebras of Hilbert space operators.

Integral Equations and Operator Theory

VX e s fA(X) = AX - XA; we denote R(~A), R(~A)- and {A}' respectively the range, the norm clo... more VX e s fA(X) = AX - XA; we denote R(~A), R(~A)- and {A}' respectively the range, the norm closure of the range and the kernel of ~A. We denote Af = {A e/:(g) : R(~A)- (2 {A*}' = {0}}. If H is finite-dimensional, Af = s If H is infinite-dimensional, this equality does not hold. So a reasonable purpose is to determine what elements are in N. When H is a separable Hilbert space, AY contains the operators A for which p(A) is normal for some quadratic polynomial p(z) [2 ],the subnormal operators with cyclic vectors [2 ] and the ison:letries [3 ]. In this paper, we show that AY contains also all the operators unitarity equivalent to Jordan operators.

Banach Journal of Mathematical Analysis, 2012

In the present paper, taking some advantages offered by the context of finite dimensional Hilbert... more In the present paper, taking some advantages offered by the context of finite dimensional Hilbert spaces, we shall give a complete characterizations of certain distinguished classes of operators (self-adjoint, unitary reflection, normal) in terms of operator inequalities. These results extend previous characterizations obtained by the second author.