• 대한수학회논문집
• /
• 제18권3호
• /
• pp.469-479
• /
• 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).

• Journal of applied mathematics & informatics
• /
• 제32권3_4호
• /
• pp.441-446
• /
• 2014

In this paper the following is proved: Let $\mathcal{L}$ be a subspace lattice on a Hilbert space $\mathcal{H}$ and X and Y be operators acting on $\mathcal{H}$. Then there exists a compact operator A in $Alg\mathcal{L}$ such that AX = Y if and only if ${\sup}\{\frac{{\parallel}E^{\perp}Yf{\parallel}}{{\parallel}E^{\perp}Xf{\parallel}}\;:\;f{\in}\mathcal{H},\;E{\in}\mathcal{L}\}$ = K < ${\infty}$ and Y is compact. Moreover, if the necessary condition holds, then we may choose an operator A such that AX = Y and ${\parallel}A{\parallel}=K$.

• 호남수학학술지
• /
• 제30권2호
• /
• pp.329-334
• /
• 2008

Given operators X and Y acting on a Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that AX = Y. In this article, the following is proved: Let $\mathcal{L}$ be a subspace lattice on $\mathcal{H}$ and let X and Y be operators acting on a Hilbert space H. Let P be the projection onto the $\overline{rangeX}$. If PE = EP for each E ${\in}$ $\mathcal{L}$, then the following are equivalent: (1) sup ${{\frac{{\parallel}E^{\perp}Yf{\parallel}}{{\parallel}E^{\perp}Xf{\parallel}}}:f{\in}\mathcal{H},\;E{\in}\mathcal{L}}$ < ${\infty},\;\overline{rangeY}\;{\subset}\;\overline{rangeX}$, and there is a bounded operator T acting on $\mathcal{H}$ such that < Xf, Tg >=< Yf, Xg >, < Tf, Tg >=< Yf, Yg > for all f and gin $\mathcal{H}$ and $T^*h$ = 0 for h ${\in}\;{\overline{rangeX}}^{\perp}$. (2) There is a normal operator A in AlgL such that AX = Y and Ag = 0 for all g in range ${\overline{rangeX}}^{\perp}$.

• Kyungpook Mathematical Journal
• /
• 제45권2호
• /
• pp.293-299
• /
• 2005

Let ${\cal{L}}$ be a commutative subspace lattice on a Hilbert space ${\cal{H}}$ and X and Y be operators on ${\cal{H}}$. Let $${\cal{M}}_X=\{{\sum}{\limits_{i=1}^n}E_{i}Xf_{i}:n{\in}{\mathbb{N}},f_{i}{\in}{\cal{H}}\;and\;E_{i}{\in}{\cal{L}}\}$$ and $${\cal{M}}_Y=\{{\sum}{\limits_{i=1}^n}E_{i}Yf_{i}:n{\in}{\mathbb{N}},f_{i}{\in}{\cal{H}}\;and\;E_{i}{\in}{\cal{L}}\}.$$ Then the following are equivalent. (i) There is an operator A in $Alg{\cal{L}}$ such that AX = Y, Ag = 0 for all g in ${\overline{{\cal{M}}_X}}^{\bot},A^*A=AA^*$ and every E in ${\cal{L}}$ reduces A. (ii) ${\sup}\;\{K(E, f)\;:\;n\;{\in}\;{\mathbb{N}},f_i\;{\in}\;{\cal{H}}\;and\;E_i\;{\in}\;{\cal{L}}\}\;<\;\infty,\;{\overline{{\cal{M}}_Y}}\;{\subset}\;{\overline{{\cal{M}}_X}}$and there is an operator T acting on ${\cal{H}}$ such that ${\langle}EX\;f,Tg{\rangle}={\langle}EY\;f,Xg{\rangle}$ and ${\langle}ET\;f,Tg{\rangle}={\langle}EY\;f,Yg{\rangle}$ for all f, g in ${\cal{H}}$ and E in ${\cal{L}}$, where $K(E,\;f)\;=\;{\parallel}{\sum{\array}{n\\i=1}}\;E_{i}Y\;f_{i}{\parallel}/{\parallel}{\sum{\array}{n\\i=1}}\;E_{i}Xf_{i}{\parallel}$.

• Journal of applied mathematics & informatics
• /
• 제14권1_2호
• /
• pp.387-395
• /
• 2004

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 n-operators satisfies the equation $AX_{i}\;=\;Y_{i}$, for i = 1,2,...,n. In this article, we showed the following: Let H be a Hilbert space and let L be a subspace lattice on H. Let X and Y be operators acting on H. Assume that range(X) is dense in H. Then the following statements are equivalent: (1) There exists an operator A in AlgL such that AX = Y, $A^{*}$ = A and every E in L reduces A. (2) sup ${\frac{$\mid$$\mid${\sum_{i=1}}^n\;E_iYf_i$\mid$$\mid$}{$\mid$$\mid${\sum_{i=1}}^n\;E_iXf_i$\mid$$\mid$}$:n{\epsilon}N,f_i{\epsilon}H\;and\;E_i{\epsilon}L}\;<\;{\infty}$ and = for all E in L and all f, g in H.

• Wind and Structures
• /
• 제21권6호
• /
• pp.693-707
• /
• 2015

The efficiency of wind turbines (WT) is primarily reflected in their ability to generate electricity at any time. Downtimes of WTs due to "conventional" inspections are cost-intensive and undesirable for investors. For this reason, there is a need for structural health monitoring (SHM) systems, to enable service and maintenance on demand and to increase the inspection intervals. In general, monitoring increases the cost effectiveness of WTs. This publication concentrates on the application of two vibration-based SHM algorithms for stability and structural change monitoring of offshore WTs. Only data driven, output-only algorithms based on stochastic subspace identification (SSI) in time domain are considered. The centerpiece of this paper deals with the rough mathematical description of the dynamic behavior of offshore WTs and with the basic presentation of stochastic subspace-based algorithms and their application to these structures. Due to the early stage of the industrial application of SHM on offshore WT on the one side and the required confidentiality to the plant manufacturer and operator on the other side, up to now it is not possible to analyze different isolated structural damages resp. changes in a systematic manner, directly by means of in-situ measurement and to make these "acknowledgements" publicly available. For this reason, the sensitivity of the methods for monitoring purposes are demonstrated through their application on long time measurements from a 1:10 large scale test rig of an offshore WT under different conditions: undamaged, different levels of loosened bolt connections between tower parts, different levels of fouling, scouring and structure inclination. The limitation and further requirements for the approaches and their applicability on real foundations are discussed along the paper.

• 호남수학학술지
• /
• 제31권3호
• /
• pp.421-428
• /
• 2009

Given operators X and Y acting on a Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that AX = Y. In this paper, we showed the following : Let $\mathcal{L}$ be a subspace lattice acting on a Hilbert space $\mathcal{H}$ and let $X_i$ and $Y_i$ be operators in B($\mathcal{H}$) for i = 1, 2, ${\cdots}$. Let $P_i$ be the projection onto $\overline{rangeX_i}$ for all i = 1, 2, ${\cdots}$. If $P_kE$ = $EP_k$ for some k in $\mathbb{N}$ and all E in $\mathcal{L}$, then the following are equivalent: (1) $sup\;\{{\frac{{\parallel}E^{\perp}({\sum}^n_{i=1}Y_if_i){\parallel}}{{\parallel}E^{\perp}({\sum}^n_{i=1}Y_if_i){\parallel}}:f{\in}H,n{\in}{\mathbb{N}},E{\in}\mathcal{L}}\}$ < ${\infty}$ range $\overline{rangeY_k}\;=\;\overline{rangeX_k}\;=\;\mathcal{H}$, and < $X_kf,\;X_kg$ >=< $Y_kf,\;Y_kg$ > for some k in $\mathbb{N}$ and for all f and g in $\mathcal{H}$. (2) There exists an operator A in Alg$\mathcal{L}$ such that $AX_i$ = $Y_i$ for i = 1, 2, ${\cdots}$ and AA$^*$ = I = A$^*$A.

• Journal of Mechanical Science and Technology
• /
• 제15권7호
• /
• pp.847-858
• /
• 2001

This paper considers the control model identification and H(sub)$\infty$ controller design for a tandem cold mill (TCM). In order to improve the performance of the existing automatic gauge control (AGC) system based on the Taylor linearized model of the TCM, a new mathematical model that can complement the Taylor linearized model is constructed by using the N4SID algorithm based on subspace method and the least squares algorithm based on ARX model. It is shown that the identified model had dynamic characteristics of the TCM than the existing Taylor linearized model. The H(sub)$\infty$ controller is designed to have robust stability to the system parameters variation, disturbance attenuation and robust tracking capability to the set-up value of strip thickness. The H(sub)$\infty$ servo problem is formulated and it is solved by using LMI (linear matrix inequality) techniques. Simulation results demonstrate the usefulness and applicability of the proposed H(sub)$\infty$ controller.

• 호남수학학술지
• /
• 제26권4호
• /
• pp.379-389
• /
• 2004

Given operators X and Y acting on a Hilbert space ${\mathcal{H}}$, an interpolating operator is a bounded operator A such that AX = Y. An interpolating operator for n-operators satisfies the equation $AX_i=Y_i$, for $i=1,2,{\cdots},n$. In this article, we obtained the following : Let ${\mathcal{H}}$ be a Hilbert space and let ${\mathcal{L}}$ be a commutative subspace lattice on ${\mathcal{H}}$. Let X and Y be operators acting on ${\mathcal{H}}$. Then the following statements are equivalent. (1) There exists an operator A in $Alg{\mathcal{L}}$ such that AX = Y, A is positive and every E in ${\mathcal{L}}$ reduces A. (2) sup ${\frac{{\parallel}{\sum}^n_{i=1}\;E_iY\;f_i{\parallel}}{{\parallel}{\sum}^n_{i=1}\;E_iX\;f_i{\parallel}}}:n{\in}{\mathbb{N}},\;E_i{\in}{\mathcal{L}}$ and $f_i{\in}{\mathcal{H}}<{\infty}$ and <${\sum}^n_{i=1}\;E_iY\;f_i$, ${\sum}^n_{i=1}\;E_iX\;f_i>\;{\geq}0$, $n{\in}{\mathbb{N}}$, $E_i{\in}{\mathcal{L}}$ and $f_i{\in}H$.

• 호남수학학술지
• /
• 제31권2호
• /
• pp.259-265
• /
• 2009

Let $\mathcal{L}$ be a subspace lattice on a Hilbert space $\mathcal{H}$. Let x and y be vectors in $\mathcal{H}$ and let $P_x$ be the projection onto sp(x). If $P_xE$ = $EP_x$ for each E ${\in}\;\mathcal{L}$, then the following are equivalent. (1) There exists an operator A in Alg$\mathcal{L}$ such that Ax = y, Af = 0 for all f in $sp(x)^{\perp}$ and A ${\geq}$ 0. (2) sup ${\frac{{\parallel}E^{\perp}y{\parallel}}{{\parallel}E^{\perp}x{\parallel}}:E{\in}\mathcal{L}}$ < ${\infty}$ < x, y > ${\geq}$ 0. Let X and Y be operators in $\mathcal{B}(\mathcal{H})$. Let P be the projection onto $\overline{rangeX}$. If PE = EP for each E ${\in}\;\mathcal{L}$, then the following are equivalent: (1) sup ${\frac{{\parallel}E^{\perp}Yf{\parallel}}{{\parallel}E^{\perp}Xf{\parallel}}:f{\in}\mathcal{H},E{\in}\mathcal{L}}$ < ${\infty}$ and < Xf, Yf > ${\geq}$ 0 for all f in H. (2) There exists a positive operator A in Alg$\mathcal{L}$ such that AX = Y.