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.
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 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
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.
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.
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.
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.
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 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
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.
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.
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.
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Papers by Ameur Seddik
of bounded linear operators acting on a complex separable Hilbert space in terms of
operator inequalities related to the arithmetic–geometric mean inequality.
of bounded linear operators acting on a complex separable Hilbert space in terms of
operator inequalities related to the arithmetic–geometric mean inequality.