Inspired by recent works on m-isometric and n-symmetric multivariables operators on Hilbert space... more Inspired by recent works on m-isometric and n-symmetric multivariables operators on Hilbert spaces, in this paper we introduce the class of (m, n)-isosymmetric multivariables operators. This new class of operators emerges as a generalization of the misometric and n-isosymmetric multioperators. We study this class of operators and give some of their basic properties. In particular, we show that if R ∈ B (d) (H) is an (m, n)isosymmetric multioperators and Q ∈ B (d) (H) is an q-nilpotent multioperators, then R + Q is an (m + 2q -2, n + 2q -1)-isosymmetric multioperators under suitable conditions. Moreover, we give some results about the joint approximate spectrum of an (m, n)-isosymmetric multioperators.
Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional c... more Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. For A = (A 1 , A 2 ...An) and B = (B 1 , B 2 ...Bn) n-tuples in B(H), we define the elementary operator ∆ A,B X : for all X ∈ I (proper bilateral ideal) and for all T ∈ ker(∆ A,B | I).
which is a further generalization of normal operator. In this paper, we give a sufficient conditi... more which is a further generalization of normal operator. In this paper, we give a sufficient condition for an injective squarep -quasihyponormal operator to be self-adjoint, and we obtain that every squarep -quasihyponormal operator has a scalar extension. As a consequence, we prove that if T is a quasiaffine transform of squarep -quasihyponormal, then T satisfies Weyl's theorem. Finally some examples are presented.
Journal of Inequalities in Pure & Applied Mathematics, 2006
The general problem in this paper is minimizing the Cp norm of suitable affine mappings from B(H)... more The general problem in this paper is minimizing the Cp norm of suitable affine mappings from B(H) to Cp, using convex and differential analysis (Gateaux derivative) as well as input from operator theory. The mappings considered generalize the so-called elementary operators and in particular the generalized derivations, which are of great interest by themselves. The main results obtained characterize global minima in terms of (Banach space) orthogonality, and constitute an interesting combination of infinite-dimensional differential analysis, operator theory and duality. Note that the results obtained generalize all results in the literature concerning operator which are orthogonal to the range of a derivation and the techniques used have not been done by other authors.
Mathematical Inequalities & Applications, 2013
In this paper we define the p − * − paranormal operators and we show some properties of this clas... more In this paper we define the p − * − paranormal operators and we show some properties of this class of operators. We also prove that a p − * − paranormal operator is polaroid and we show a necessary and sufficient condition for the Riesz idempotent associated to a non-zero isolated point of the spectrum of a p − * − paranormal operator to be self-adjoint. Finally, we show that generalized a-Weyl's theorem holds for p − * − paranormal operators and we present some finite operators.
m -symmetric operator plays a crucial role in the development of operator theory and has been wid... more m -symmetric operator plays a crucial role in the development of operator theory and has been widely studied due to unexpected intimate connections with differential equations, particularly conjugate point theory and disconjugacy. For positive integers m and k , an operator T is said to be a k -quasim -symmetric operator if , which is a generalization of m -symmetric operator. In this paper, some basic structural properties of k -quasim -symmetric operators are established with the help of operator matrix representation. In particular, we also show that every k -quasi-3 -symmetric operator has a scalar extension. Finally, we prove that generalized Weyl's theorem holds for k -quasi-3 -symmetric operators.
Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra ... more Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H. Let A, B ∈ L(H). Define the generalized derivation δA,B : L(H) 7→ L(H) by δA,B(X) = AX − XB. The class of generalized P symmetric operators is the class of all pairs of operators A, B ∈ L(H) such that [TA = BT implies A∗T = TB∗, T ∈ C1(H)] (*) (trace class operators), i.e, the pair (A, B) satisfies the FugledePutnam’s theorem in C1(H). In this paper we present new C ∗-algebras generated by the pair (A, B) satisfying (*). Other related results are also given.
Abstract. If S,T ∈ B(H) have Bishop’s property (β), does S +T have Bishop’s property (β)? In this... more Abstract. If S,T ∈ B(H) have Bishop’s property (β), does S +T have Bishop’s property (β)? In this paper, a special case of this question is studied. Also given are a necessary and sufficient condition for a 2×2 operator matrix to have Bishop’s property (β). Finally, the Helton class of an operator which has Bishop’s property (β) is studied.
Let T be a bounded linear operator on a complex Hilbert space H. In this paper we investigate som... more Let T be a bounded linear operator on a complex Hilbert space H. In this paper we investigate some, properties of the class of n-power quasinormal operators , denoted [nQN ], satisfying T n |T | 2 − |T | 2 T n = 0 and some relations between n-normal operators and n-quasinormal operators.
The aim of this paper is to present certain basic properties of some classes of nonnormal operato... more The aim of this paper is to present certain basic properties of some classes of nonnormal operators defined on a complex separable Hilbert space. Both of the normality of their integer powers and their relations with isometries are established. The ascent of such operators as well as other important related results are also established. The decomposition of such operators, their restrictions on invariant subspaces, and some spectral properties are also presented.
Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra ... more Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H into itself. Let A, B ∈ L(H) we define the generalized derivation δA,B : L(H) → L(H) by δA,B(X) = AX − XB, we note δA,A = δA. If the inequality ||T − (AX − XA)|| ≥ ||T || holds for all X ∈ L(H) and for all T ∈ kerδA, then we say that the range of δA is orthogonal to the kernel of δA in the sense of Birkhoff. The operator A ∈ L(H) is said to be finite [22] if ||I − (AX − XA)|| ≥ 1(*) for all X ∈ L(H), where I is the identity operator. The well-known inequality (*), due to J. P. Williams [22] is the starting point of the topic of commutator approximation (a topic which has its roots in quantum theory [23]). In [16], the author showed that a paranormal operator is finite. In this paper we present some new classes of finite operators containing the class of paranormal operators and we prove that the range of a generalized derivation is orthogonal to its kernel for a large class of operators containing the class of normal operators.
We prove the orthogonality (in the sense of Birkhoff) of the range and the kernel of an important... more We prove the orthogonality (in the sense of Birkhoff) of the range and the kernel of an important class of elementary operators with respect to the Schatten p-class.
Let A be a bounded linear operator acting on a Hilbert space H. In [32], A. Uchiyama proved that ... more Let A be a bounded linear operator acting on a Hilbert space H. In [32], A. Uchiyama proved that Weyl's theorem holds for class A operators with the additional condition that ker A| [T H] = 0 and he showed that every class A operator whose Weyl spectrum equals to zero is compact and normal. In this paper we show that Weyl's theorem holds for algebraically class A operator without the additional condition ker A| [T H] = 0. This leads as to show that a class A operator whose Weyl spectrum equals to zero is always compact and normal.
We give some results concerning the orthogonality of the range and the kernel of a gener-alized d... more We give some results concerning the orthogonality of the range and the kernel of a gener-alized derivation modulo the ideal of all compact operators. 2000 Mathematics Subject Classification: 47B47, 47A30, 47B20, 47B10. 1. Introduction. Let
We are interested in the investigation of the orthogonality (in the sense of Birkhoff) of the ran... more We are interested in the investigation of the orthogonality (in the sense of Birkhoff) of the range of an elementary operator and its kernel.
ABSTRACT. The general problem in this paper is minimizing the Cp − norm of suitable affine mappin... more ABSTRACT. The general problem in this paper is minimizing the Cp − norm of suitable affine mappings from B(H) to Cp, using convex and differential analysis (Gateaux derivative) as well as input from operator theory. The mappings considered generalize the so-called elementary operators and in particular the generalized derivations, which are of great interest by themselves. The main results obtained characterize global minima in terms of (Banach space) orthogonality, and constitute an interesting combination of infinite-dimensional differential analysis, operator theory and duality. Note that the results obtained generalize all results in the literature concerning operator which are orthogonal to the range of a derivation and the techniques used have not been done by other authors.
Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra ... more Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators onH. Recently H.Kim proved that Fuglede-Putnam’s theorem holds for injective (p, k)-quasihyponormal and p-hyponormal operators. In this paper we prove that Fuglede-Putnam’s theorem holds for injective (p, k)-quasihyponormal and log-hyponormal operators. We also show that this results remains true for dominant and injective (p, k)-quasihyponormal operators. Other related results are also given.
Journal of Inequalities in Pure & Applied Mathematics, 2005
This note studies how certain problems in quantum theory have motivated some recent research in p... more This note studies how certain problems in quantum theory have motivated some recent research in pure Mathematics in matrix and operator theory. The mathematical key is that of a commutator. We introduce the notion of the pair (A,B) of operators having the Fuglede- Putnam's property in the ideal of all compact operators. The characterization of this class leads us to generalize some recent results. We also give some applications of these results.
Inspired by recent works on m-isometric and n-symmetric multivariables operators on Hilbert space... more Inspired by recent works on m-isometric and n-symmetric multivariables operators on Hilbert spaces, in this paper we introduce the class of (m, n)-isosymmetric multivariables operators. This new class of operators emerges as a generalization of the misometric and n-isosymmetric multioperators. We study this class of operators and give some of their basic properties. In particular, we show that if R ∈ B (d) (H) is an (m, n)isosymmetric multioperators and Q ∈ B (d) (H) is an q-nilpotent multioperators, then R + Q is an (m + 2q -2, n + 2q -1)-isosymmetric multioperators under suitable conditions. Moreover, we give some results about the joint approximate spectrum of an (m, n)-isosymmetric multioperators.
Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional c... more Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. For A = (A 1 , A 2 ...An) and B = (B 1 , B 2 ...Bn) n-tuples in B(H), we define the elementary operator ∆ A,B X : for all X ∈ I (proper bilateral ideal) and for all T ∈ ker(∆ A,B | I).
which is a further generalization of normal operator. In this paper, we give a sufficient conditi... more which is a further generalization of normal operator. In this paper, we give a sufficient condition for an injective squarep -quasihyponormal operator to be self-adjoint, and we obtain that every squarep -quasihyponormal operator has a scalar extension. As a consequence, we prove that if T is a quasiaffine transform of squarep -quasihyponormal, then T satisfies Weyl's theorem. Finally some examples are presented.
Journal of Inequalities in Pure & Applied Mathematics, 2006
The general problem in this paper is minimizing the Cp norm of suitable affine mappings from B(H)... more The general problem in this paper is minimizing the Cp norm of suitable affine mappings from B(H) to Cp, using convex and differential analysis (Gateaux derivative) as well as input from operator theory. The mappings considered generalize the so-called elementary operators and in particular the generalized derivations, which are of great interest by themselves. The main results obtained characterize global minima in terms of (Banach space) orthogonality, and constitute an interesting combination of infinite-dimensional differential analysis, operator theory and duality. Note that the results obtained generalize all results in the literature concerning operator which are orthogonal to the range of a derivation and the techniques used have not been done by other authors.
Mathematical Inequalities & Applications, 2013
In this paper we define the p − * − paranormal operators and we show some properties of this clas... more In this paper we define the p − * − paranormal operators and we show some properties of this class of operators. We also prove that a p − * − paranormal operator is polaroid and we show a necessary and sufficient condition for the Riesz idempotent associated to a non-zero isolated point of the spectrum of a p − * − paranormal operator to be self-adjoint. Finally, we show that generalized a-Weyl's theorem holds for p − * − paranormal operators and we present some finite operators.
m -symmetric operator plays a crucial role in the development of operator theory and has been wid... more m -symmetric operator plays a crucial role in the development of operator theory and has been widely studied due to unexpected intimate connections with differential equations, particularly conjugate point theory and disconjugacy. For positive integers m and k , an operator T is said to be a k -quasim -symmetric operator if , which is a generalization of m -symmetric operator. In this paper, some basic structural properties of k -quasim -symmetric operators are established with the help of operator matrix representation. In particular, we also show that every k -quasi-3 -symmetric operator has a scalar extension. Finally, we prove that generalized Weyl's theorem holds for k -quasi-3 -symmetric operators.
Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra ... more Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H. Let A, B ∈ L(H). Define the generalized derivation δA,B : L(H) 7→ L(H) by δA,B(X) = AX − XB. The class of generalized P symmetric operators is the class of all pairs of operators A, B ∈ L(H) such that [TA = BT implies A∗T = TB∗, T ∈ C1(H)] (*) (trace class operators), i.e, the pair (A, B) satisfies the FugledePutnam’s theorem in C1(H). In this paper we present new C ∗-algebras generated by the pair (A, B) satisfying (*). Other related results are also given.
Abstract. If S,T ∈ B(H) have Bishop’s property (β), does S +T have Bishop’s property (β)? In this... more Abstract. If S,T ∈ B(H) have Bishop’s property (β), does S +T have Bishop’s property (β)? In this paper, a special case of this question is studied. Also given are a necessary and sufficient condition for a 2×2 operator matrix to have Bishop’s property (β). Finally, the Helton class of an operator which has Bishop’s property (β) is studied.
Let T be a bounded linear operator on a complex Hilbert space H. In this paper we investigate som... more Let T be a bounded linear operator on a complex Hilbert space H. In this paper we investigate some, properties of the class of n-power quasinormal operators , denoted [nQN ], satisfying T n |T | 2 − |T | 2 T n = 0 and some relations between n-normal operators and n-quasinormal operators.
The aim of this paper is to present certain basic properties of some classes of nonnormal operato... more The aim of this paper is to present certain basic properties of some classes of nonnormal operators defined on a complex separable Hilbert space. Both of the normality of their integer powers and their relations with isometries are established. The ascent of such operators as well as other important related results are also established. The decomposition of such operators, their restrictions on invariant subspaces, and some spectral properties are also presented.
Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra ... more Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H into itself. Let A, B ∈ L(H) we define the generalized derivation δA,B : L(H) → L(H) by δA,B(X) = AX − XB, we note δA,A = δA. If the inequality ||T − (AX − XA)|| ≥ ||T || holds for all X ∈ L(H) and for all T ∈ kerδA, then we say that the range of δA is orthogonal to the kernel of δA in the sense of Birkhoff. The operator A ∈ L(H) is said to be finite [22] if ||I − (AX − XA)|| ≥ 1(*) for all X ∈ L(H), where I is the identity operator. The well-known inequality (*), due to J. P. Williams [22] is the starting point of the topic of commutator approximation (a topic which has its roots in quantum theory [23]). In [16], the author showed that a paranormal operator is finite. In this paper we present some new classes of finite operators containing the class of paranormal operators and we prove that the range of a generalized derivation is orthogonal to its kernel for a large class of operators containing the class of normal operators.
We prove the orthogonality (in the sense of Birkhoff) of the range and the kernel of an important... more We prove the orthogonality (in the sense of Birkhoff) of the range and the kernel of an important class of elementary operators with respect to the Schatten p-class.
Let A be a bounded linear operator acting on a Hilbert space H. In [32], A. Uchiyama proved that ... more Let A be a bounded linear operator acting on a Hilbert space H. In [32], A. Uchiyama proved that Weyl's theorem holds for class A operators with the additional condition that ker A| [T H] = 0 and he showed that every class A operator whose Weyl spectrum equals to zero is compact and normal. In this paper we show that Weyl's theorem holds for algebraically class A operator without the additional condition ker A| [T H] = 0. This leads as to show that a class A operator whose Weyl spectrum equals to zero is always compact and normal.
We give some results concerning the orthogonality of the range and the kernel of a gener-alized d... more We give some results concerning the orthogonality of the range and the kernel of a gener-alized derivation modulo the ideal of all compact operators. 2000 Mathematics Subject Classification: 47B47, 47A30, 47B20, 47B10. 1. Introduction. Let
We are interested in the investigation of the orthogonality (in the sense of Birkhoff) of the ran... more We are interested in the investigation of the orthogonality (in the sense of Birkhoff) of the range of an elementary operator and its kernel.
ABSTRACT. The general problem in this paper is minimizing the Cp − norm of suitable affine mappin... more ABSTRACT. The general problem in this paper is minimizing the Cp − norm of suitable affine mappings from B(H) to Cp, using convex and differential analysis (Gateaux derivative) as well as input from operator theory. The mappings considered generalize the so-called elementary operators and in particular the generalized derivations, which are of great interest by themselves. The main results obtained characterize global minima in terms of (Banach space) orthogonality, and constitute an interesting combination of infinite-dimensional differential analysis, operator theory and duality. Note that the results obtained generalize all results in the literature concerning operator which are orthogonal to the range of a derivation and the techniques used have not been done by other authors.
Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra ... more Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators onH. Recently H.Kim proved that Fuglede-Putnam’s theorem holds for injective (p, k)-quasihyponormal and p-hyponormal operators. In this paper we prove that Fuglede-Putnam’s theorem holds for injective (p, k)-quasihyponormal and log-hyponormal operators. We also show that this results remains true for dominant and injective (p, k)-quasihyponormal operators. Other related results are also given.
Journal of Inequalities in Pure & Applied Mathematics, 2005
This note studies how certain problems in quantum theory have motivated some recent research in p... more This note studies how certain problems in quantum theory have motivated some recent research in pure Mathematics in matrix and operator theory. The mathematical key is that of a commutator. We introduce the notion of the pair (A,B) of operators having the Fuglede- Putnam's property in the ideal of all compact operators. The characterization of this class leads us to generalize some recent results. We also give some applications of these results.
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