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ANALYTIC EXTENSIONS OF M-HYPONORMAL OPERATORS
MECHERI, SALAH,ZUO, FEI Korean Mathematical Society 2016 대한수학회지 Vol.53 No.1
In this paper, we introduce the class of analytic extensions of M-hyponormal operators and we study various properties of this class. We also use a special Sobolev space to show that every analytic extension of an M-hyponormal operator T is subscalar of order 2k + 2. Finally we obtain that an analytic extension of an M-hyponormal operator satisfies Weyl's theorem.
Range Kernel Orthogonality and Finite Operators
Mecheri, Salah,Abdelatif, Toualbia Department of Mathematics 2015 Kyungpook mathematical journal Vol.55 No.1
Let H be a separable infinite dimensional complex Hilbert space, and let $\mathcal{L}(H)$ denote the algebra of all bounded linear operators on H into itself. Let $A,B{\in}\mathcal{L}(H)$ we define the generalized derivation ${\delta}_{A,B}:\mathcal{L}(H){\mapsto}\mathcal{L}(H)$ by ${\delta}_{A,B}(X)=AX-XB$, we note ${\delta}_{A,A}={\delta}_A$. If the inequality ${\parallel}T-(AX-XA){\parallel}{\geq}{\parallel}T{\parallel}$ holds for all $X{\in}\mathcal{L}(H)$ and for all $T{\in}ker{\delta}_A$, then we say that the range of ${\delta}_A$ is orthogonal to the kernel of ${\delta}_A$ in the sense of Birkhoff. The operator $A{\in}\mathcal{L}(H)$ is said to be finite [22] if ${\parallel}I-(AX-XA){\parallel}{\geq}1(*)$ for all $X{\in}\mathcal{L}(H)$, 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.
Positive answer to the conjecture by Fong and Istratescu
Salah Mecheri,Ahmed Bachir 대한수학회 2005 대한수학회보 Vol.42 No.4
In this note we give a positive answer to the conjectureby Fong and Istratescu.
Spectral Properties of k-quasi-class A(s, t) Operators
Mecheri, Salah,Braha, Naim Latif Department of Mathematics 2019 Kyungpook mathematical journal Vol.59 No.3
In this paper we introduce a new class of operators which will be called the class of k-quasi-class A(s, t) operators. An operator $T{\in}B(H)$ is said to be k-quasi-class A(s, t) if $$T^{*k}(({\mid}T^*{\mid}^t{\mid}T{\mid}^{2s}{\mid}T^*{\mid}^t)^{\frac{1}{t+s}}-{\mid}T^*{\mid}^{2t})T^k{\geq}0$$, where s > 0, t > 0 and k is a natural number. We show that an algebraically k-quasi-class A(s, t) operator T is polaroid, has Bishop's property ${\beta}$ and we prove that Weyl type theorems for k-quasi-class A(s, t) operators. In particular, we prove that if $T^*$ is algebraically k-quasi-class A(s, t), then the generalized a-Weyl's theorem holds for T. Using these results we show that $T^*$ satisfies generalized the Weyl's theorem if and only if T satisfies the generalized Weyl's theorem if and only if T satisfies Weyl's theorem. We also examine the hyperinvariant subspace problem for k-quasi-class A(s, t) operators.
Fuglede-Putnam theorem for $p$-hyponormal or class ${\mathcal Y}$
Salah Mecheri,Kotaro Tanahashi,Atsushi Uchiyama 대한수학회 2006 대한수학회보 Vol.43 No.4
We say operatorsA;Bon Hilbert space satisfy Fuglede-Putnam theorem ifAX = XB for someX implies AX = XB.We show that if either (1)A is p-hyponormal and Bis a classYoperator or (2)A is a classY operator and Bis p-hyponormal,then A;B satisfy Fuglede-Putnam theorem.
POSITIVE ANSWER TO THE CONJECTURE BY FONG AND ISTRATESCU
MECHERI SALAH,BACHIR AHMED Korean Mathematical Society 2005 대한수학회보 Vol.42 No.4
In this note we give a positive answer to the conjecture by Fong and Istratescu.
Analytic extensions of $M$-hyponormal operators
Salah Mecheri,Fei Zuo 대한수학회 2016 대한수학회지 Vol.53 No.1
In this paper, we introduce the class of analytic extensions of $M$-hyponormal operators and we study various properties of this class. We also use a special Sobolev space to show that every analytic extension of an $M$-hyponormal operator $T$ is subscalar of order $2k+2$. Finally we obtain that an analytic extension of an $M$-hyponormal operator satisfies Weyl's theorem.
FUGLEDE-PUTNAM THEOREM FOR p-HYPONORMAL OR CLASS y OPERATORS
Mecheri, Salah,Tanahashi, Kotaro,Uchiyama, Atsushi Korean Mathematical Society 2006 대한수학회보 Vol.43 No.4
We say operators A, B on Hilbert space satisfy Fuglede-Putnam theorem if AX = X B for some X implies $A^*X=XB^*$. We show that if either (1) A is p-hyponormal and $B^*$ is a class y operator or (2) A is a class y operator and $B^*$ is p-hyponormal, then A, B satisfy Fuglede-Putnam theorem.
Generalized Weyl’s Theorem for Some Classes of Operators
MECHERI, SALAH 대한수학회 2006 Kyungpook mathematical journal Vol.46 No.4
Let A be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of A is the set _(σBω)(A) of all λ ∈ C such that A-λI is not a B-Fredholm operator of index 0. Let E(A) be the set of all isolated eigenvalues of A. Recently in [6] Berkani showed that if A is a hyponormal operator, then A satisfies generalized Weyl's theorem _(σBω)(A) =σ(A) \ E(A), and the B-Weyl spectrum _(σBω)(A) of A satisfies the spectral mapping theorem. In [51], H. Weyl proved that weyl's theorem holds for hermitian operators. Weyl's theorem has been extended from hermitian operators to hyponormal and Toeplitz operators [12], and to several classes of operators including semi-normal operators ([9], [10]). Recently W. Y. Lee [35] showed that Weyl's theorem holds for algebraically hyponormal operators. R. Curto and Y. M. Han [14] have extended Lee's results to algebraically paranormal operators. In [19] the authors showed that Weyl's theorem holds for algebraically p-hyponormal operators. As Berkani has shown in [5], if the generalized Weyl's theorem holds for A, then so does Weyl's theorem. In this paper all the above results are generalized by proving that generalized Weyl's theorem holds for the case where A is an algebraically (p, k)-quasihyponormal or an algebarically paranormal operator which includes all the above mentioned operators.
Some Properties of (y) Class Operators
Bachir, Ahmed,Mecheri, Salah Department of Mathematics 2009 Kyungpook mathematical journal Vol.49 No.2
In this paper we study some spectral properties of the class (y) operators and we will investigate on the relation between this class and other usual classes of operators.