• Korean Journal of Mathematics
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• 제21권1호
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• pp.75-80
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• 2013

Let T be a bounded linear operator on a complex Hilbert space $\mathfrak{H}$. An operator T is called class A operator if ${\mid}T^2{\mid}{\geq}{\mid}T{\mid}^2$ and is called class $A({\kappa})$ operator if $(T^*{\mid}T{\mid}^{2{\kappa}}T)^{\frac{1}{{\kappa}+1}}{\geq}{\mid}T{\mid}^2$ for a positive number ${\kappa}$. In this paper, we show that ${\sigma}$ is continuous when restricted to the set of class $A({\kappa})$ operators.

• 대한수학회논문집
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• 제17권3호
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• pp.487-493
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• 2002

Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. An interpolating operator for the n-operators satisfies the equation AX$\_$i/ : Y$\_$i/, for i = 1, 2 …, n. In this article, we obtained the following : Let X = (x$\_$ij/) and Y = (y$\_$ij/) be operators acting on H such that $\varkappa$$\_$ i$\sigma$ (i)/ 0 for all i. Then the following statements are equivalent. (1) There exists a unitary operator A in Alg(equation omitted) such that AX = Y and every E in (equation omitted) reduces A. (2) sup{(equation omitted)}<$\infty$ and (equation omitted) = 1 for all i = 1, 2, ….

• 대한수학회보
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• 제22권1호
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• pp.23-29
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• 1985

Let H be an arbitrary complex Hilbert space. We constructed an extension K of H by means of weakly convergent sequences in H and the Banach limit. Let .phi. be the faithful *-representation of L(H) on K. In this note, we investigated the relations between spectra of T in L(H) and .phi.(T) in L(K) and we obtained the following results: 1) If T is a compact operator on H, then .phi.(T) is also a compact operator on K (Proposition 6), 2) .sigma.$_{l}$ (.phi.(T)).contnd..sigma.$_{l}$ (T) for any operator T.mem.L(H) (Corollary 10), 3) For every operator T.mem.L(H), .sigma.$_{ap}$ (.phi.(T))=.sigma.$_{ap}$ (T))=.sigma.$_{ap}$ (T)=.sigma.$_{p}$(.phi.(T)) (Lemma 12, 13) and .sigma.$_{c}$(.phi.(T))=.sigma.(Theorem 15).15).

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제5권2호
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• pp.109-114
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• 1998

Let H be a separable complex H be a space and let (equation omitted)(H) be the *-algebra of all bounded linear operators on H. An operator T in (equation omitted)(H) is said to be p-hyponormal if ($T^{\ast}T)^p - (TT^{\ast})^{p}\geq$ 0 for 0 ＜ p ＜ 1. If p = 1, T is hyponormal and if p = $\frac{1}{2}$, T is semi-hyponormal. In this paper, by using a technique introduced by S. K. Berberian, we show that the approximate point spectrum $\sigma_{\alpha p}(T) of a pure p-hyponormal operator T is empty, and obtains the compact perturbation of T.

• 대한수학회지
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• 제55권1호
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• pp.225-251
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• 2018

The m-isometry of a single operator in Agler and Stankus [3] was naturally generalized to the m-isometric tuple of several commuting operators by Gleason and Richter [22]. Some examples of m-isometric tuples including the recently much studied Arveson-Drury d-shift were given in [22]. We provide more examples of m-isometric tuples of operators by using sums of operators or products of operators or functions of operators. A class of m-isometric tuples of unilateral weighted shifts parametrized by polynomials are also constructed. The examples in Gleason and Richter [22] are then obtained by choosing some specific polynomials. This work extends partially results obtained in several recent papers on the m-isometry of a single operator.

• 대한수학회보
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• 제54권4호
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• pp.1443-1455
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• 2017

Let $M_{z^N}(N{\in}{\mathbb{Z}}^d_+)$ be a bounded multiplication operator on a class of Hilbert spaces with orthogonal basis $\{z^n:n{\in}{\mathbb{Z}}^d_+\}$. In this paper, we prove that each reducing subspace of $M_{z^N}$ is the direct sum of some minimal reducing subspaces. For the case that d = 2, we find all the minimal reducing subspaces of $M_{z^N}$ ($N=(N_1,N_2)$, $N_1{\neq}N_2$) on weighted Bergman space $A^2_{\alpha}({\mathbb{B}}_2)$(${\alpha}$ > -1) and Hardy space $H^2({\mathbb{B}}_2)$, and characterize the structure of ${\mathcal{V}}^{\ast}(z^N)$, the commutant algebra of the von Neumann algebra generated by $M_{z^N}$.

• Kyungpook Mathematical Journal
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• 제58권1호
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• pp.81-90
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• 2018

A bounded linear operator T on a Hilbert space ${\mathcal{H}}$ is convex, if for each $x{\in}{\mathcal{H}}$, ${\parallel}T^2x{\parallel}^2-2{\parallel}Tx{\parallel}^2+{\parallel}x{\parallel}^2{\geq}0$. In this paper, it is shown that if T is convex and supercyclic then it is a contraction or an expansion. We then present some examples of convex supercyclic operators. Also, it is proved that no convex composition operator induced by an automorphism of the disc on a weighted Hardy space is supercyclic.

• Kyungpook Mathematical Journal
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• 제46권4호
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• pp.553-563
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• 2006

Let A be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of A is the set ${\sigma}_{B{\omega}}(A)$ of all ${\lambda}{\in}\mathbb{C}$ such that $A-{\lambda}I$ 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 ${\sigma}_{B{\omega}}(A)={\sigma}(A)$\E(A), and the B-Weyl spectrum ${\sigma}_{B{\omega}}(A)$ 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 generalizedWeyl'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.

• 대한수학회지
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• 제43권4호
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• pp.899-909
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• 2006

Let T be a bounded linear operator on a complex infinite dimensional Hilbert space $\scr{H}$. We say that T is a quasi-class A operator if $T^*\|T^2\|T{\geq}T^*\|T\|^2T$. In this paper we prove that if T is a quasi-class A operator and f is a function analytic on a neigh-borhood or the spectrum or T, then f(T) satisfies Weyl's theorem and f($T^*$) satisfies a-Weyl's theorem.

• 대한수학회지
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• 제41권4호
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• pp.617-627
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• 2004

Let (equation omitted) denote the class of bounded linear Hilbert space operators with the property that $\midA^2\mid\geq\midA\mid^2$. In this paper we show that (equation omitted)-operators are finitely ascensive and that, for non-zero operators A and B, A (equation omitted) B is in (equation omitted) if and only if A and B are in (equation omitted). Also, it is shown that if A is an operator such that p(A) is in (equation omitted) for a non-trivial polynomial p, then Weyl's theorem holds for f(A), where f is a function analytic on an open neighborhood of the spectrum of A.