• Communications of the Korean Mathematical Society
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• v.18 no.3
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• pp.469-479
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• 2003

Let B be the open unit ball in $C^{n}$ and ${\mu}_{q}$(q > -1) the Lebesgue measure such that ${\mu}_{q}$(B) ＝ 1. Let ${L_{a,q}}^2$ be the subspace of ${L^2(B,D{\mu}_q)$ consisting of analytic functions, and let $\overline{{L_{a,q}}^2}$ be the subspace of ${L^2(B,D{\mu}_q)$) consisting of conjugate analytic functions. Let $\bar{P}$ be the orthogonal projection from ${L^2(B,D{\mu}_q)$ into $\overline{{L_{a,q}}^2}$. The little Hankel operator ${h_{\varphi}}^{q}\;:\;{L_{a,q}}^2\;{\rightarrow}\;{\overline}{{L_{a,q}}^2}$ is defined by ${h_{\varphi}}^{q}(\cdot)\;=\;{\bar{P}}({\varphi}{\cdot})$. In this paper, we will find the necessary and sufficient condition that the little Hankel operator ${h_{\varphi}}^{q}$ is bounded(or compact).

• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.597-602
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• 1997

Let H be a separable complex Hilbert space and L(H) be the *-algebra of all bounded linear operators on H. For $T \in L(H)$, we construct a pair of semi-positive definite operators $$ $\mid$T$\mid$_r = (T^*T)^{\frac{1}{2}} and $\mid$T$\mid$_l = (TT^*)^{\frac{1}{2}}. $$ An operator T is called a semi-hyponormal operator if $$ Q_T = $\mid$T$\mid$_r - $\mid$T$\mid$_l \geq 0. $$ In this paper, by using a technique introduced by Berberian [1], we show that the approximate point spectrum $\sigma_{ap}(T)$ of a semi-hyponomal operator T is empty.

• Journal of the Korean Mathematical Society
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• v.43 no.4
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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.

• Proceedings of the KIEE Conference
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• 1994.07a
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• pp.465-467
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• 1994

A VSC with Inertial COFB(Coordinate-Operator Feedback) is presented for chattering alleviation. Athought the conventional sliding mode controller has good properties of robustness for disturbances or parameter variations, fast response, and easy implementation, there exists an inevitable chattering problem which deteriorates the control performance of system. VSC using Inertial COFB has properties of bounded feedback gain, reduced chattering, and robustness for disturbances or parameter variations. The validity of the proposed method is demonstrated through computer simulation for a position control of BLDCM.

• Bulletin of the Korean Mathematical Society
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• v.51 no.2
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• pp.423-435
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• 2014

In this paper, the fractional Hardy-type operator of variable order ${\beta}(x)$ is shown to be bounded from the Herz-Morrey spaces $M\dot{K}^{{\alpha},{\lambda}}_{p_1,q_1({\cdot})}(\mathbb{R}^n)$ with variable exponent $q_1(x)$ into the weighted space $M\dot{K}^{{\alpha},{\lambda}}_{p_2,q_2({\cdot})}(\mathbb{R}^n,{\omega})$, where ${\omega}=(1+|x|)^{-{\gamma}(x)}$ with some ${\gamma}(x)$ > 0 and $1/q_1(x)-1/q_2(x)={\beta}(x)/n$ when $q_1(x)$ is not necessarily constant at infinity. It is assumed that the exponent $q_1(x)$ satisfies the logarithmic continuity condition both locally and at infinity that 1 < $q_1({\infty}){\leq}q_1(x){\leq}(q_1)+$ < ${\infty}(x{\in}\mathbb{R}^n)$.

• Journal of the Korean Mathematical Society
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• v.41 no.4
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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.

• Honam Mathematical Journal
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• v.29 no.1
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• pp.55-60
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• 2007

Given operators X and Y acting on a Hilbert space $\cal{H}$, an interpolating operator is a bounded operator A such that AX = Y. In this article, we showed the following : Let $\cal{L}$ be a subspace lattice acting on a Hilbert space $\cal{H}$ and let X and Y be operators in $\cal{B}(\cal{H})$. Let P be the projection onto $\bar{rangeX}$. If FE = EF for every $E\in\cal{L}$, then the following are equivalent: (1) $sup\{{{\parallel}E^{\perp}Yf\parallel\atop \parallel{E}^{\perp}Xf\parallel}\;:\;f{\in}\cal{H},\;E\in\cal{L}\}\$ < $\infty$, $\bar{range\;Y}\subset\bar{range\;X}$, and < Xf, Yg >=< Yf,Xg > for any f and g in $\cal{H}$. (2) There exists a self-adjoint operator A in Alg$\cal{L}$ such that AX = Y.

• Honam Mathematical Journal
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• v.29 no.4
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• pp.523-533
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• 2007

Let $\cal{B}$ be a reflexive Banach space of functions analytic on the open unit disc and M be an invariant subspace of the multiplication operator by the independent variable, $M_z$. Suppose that $\varphi\;\in\;\cal{H}^{\infty}$ and $M_{\varphi}$ : M ${\rightarrow}$ M, defined by $M_{\varphi}f={\varphi}f$, is the operator of multiplication by ${\varphi}$. We would like to investigate the spectrum and the essential spectrum of $M_{\varphi}$ and we are looking for the necessary and sufficient conditions for $M_{\varphi}$ to be a Fredholm operator. Also we give a sufficient condition for a sequence $\{w_n\}$ to be an interpolating sequence for $\cal{B}$. At last the commutant of $M_{\varphi}$ under certain conditions on M and ${\varphi}$ is determined.

• Journal of applied mathematics & informatics
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• v.19 no.1_2
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• pp.447-452
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• 2005

Given operators X and Y on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. In this article, we investigate compact interpolation problems for vectors in a tridiagonal algebra. Let L be a subspace lattice acting on a separable complex Hilbert space H and Alg L be a tridiagonal algebra. Let X = $(x_{ij})\;and\;Y\;=\;(y_{ij})$ be operators acting on H. Then the following are equivalent: (1) There exists a compact operator A = $(x_{ij})$ in AlgL such that AX = Y. (2) There is a sequence {$\alpha_n$} in $\mathbb{C}$ such that {$\alpha_n$} converges to zero and $$y_1\;_j=\alpha_1x_1\;_j+\alpha_2x_2\;_j\;y_{2k}\;_j=\alpha_{4k-1}x_{2k\;j}\;y_{2k+1\;j}=\alpha_{4k}x_{2k\;j}+\alpha_{4k+1}x_{2k+1\;j}+\alpha_{4k+2}x_{2k+2\;j\;for\;all\;k\;\epsilon\;\mathbb{N}$$.

• Korean Journal of Mathematics
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• v.16 no.4
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• pp.583-587
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• 2008

Let $\mathfrak{L(H)}$ denote the algebra of bounded linear operators on a separable infinite dimensional complex Hilbert space $\mathfrak{H}$. We say that $T{\in}\mathfrak{L(H)}$ is a quasi-class A operator if $$T^*{\mid}T^2{\mid}T{{\geq}}T^*{\mid}T{\mid}^2T$$. In this paper we prove that if A and B are quasi-class A operators, and $B^*$ is invertible, then for a Hilbert-Schmidt operator X $$AX=XB\;implies\;A^*X=XB^*$$.