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SELF-ADJOINT INTERPOLATION ON AX = Y IN $\mathcal{B}(\mathcal{H})$
Kwak, Sung-Kon,Kim, Ki-Sook The Honam Mathematical Society 2008 호남수학학술지 Vol.30 No.4
Given operators $X_i$ and $Y_i$ (i = 1, 2, ${\cdots}$, n) acting on a Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A acting on $\mathcal{H}$ such that $AX_i$ = $Y_i$ for i= 1, 2, ${\cdots}$, n. In this article, if the range of $X_k$ is dense in H for a certain k in {1, 2, ${\cdots}$, n), then the following are equivalent: (1) There exists a self-adjoint operator A in $\mathcal{B}(\mathcal{H})$ stich that $AX_i$ = $Y_i$ for I = 1, 2, ${\cdots}$, n. (2) $sup\{{\frac{{\parallel}{\sum}^n_{i=1}Y_if_i{\parallel}}{{\parallel}{\sum}^n_{i=1}X_if_i{\parallel}}:f_i{\in}H}\}$ < ${\infty}$ and < $X_kf,Y_kg$ >=< $Y_kf,X_kg$> for all f, g in $\mathcal{H}$.
Kwak, Jongheon,Han, Sung Hyun,Moon, Hong Chul,Kim, Jin Kon,Koo, Jaseung,Lee, Jeong-Soo,Pryamitsyn, Victor,Ganesan, Venkat American Chemical Society 2015 Macromolecules Vol.48 No.4
<P><B>Graphic Abstract</B> <IMG SRC='http://pubs.acs.org/appl/literatum/publisher/achs/journals/content/mamobx/2015/mamobx.2015.48.issue-4/ma502192k/production/images/medium/ma-2014-02192k_0006.gif'></P><P><A href='http://pubs.acs.org/doi/suppl/10.1021/ma502192k'>ACS Electronic Supporting Info</A></P>
INVERTIBLE INTERPOLATION ON Ax = y IN A TRIDIAGONAL ALGEBRA ALGℒ
Kwak, Sung-Kon,Kang, Joo-Ho The Honam Mathematical Society 2011 호남수학학술지 Vol.33 No.1
Given vectors x and y in a separable complex Hilbert space $\cal{H}$, an interpolating operator is a bounded operator A such that Ax = y. We show the following : Let Alg$\cal{L}$ be a tridiagonal algebra on a separable complex Hilbert space H and let x = ($x_i$) and y = ($y_i$) be vectors in H. Then the following are equivalent: (1) There exists an invertible operator A = ($a_{kj}$) in Alg$\cal{L}$ such that Ax = y. (2) There exist bounded sequences $\{{\alpha}_n\}$ and $\{{{\beta}}_n\}$ in $\mathbb{C}$ such that for all $k\in\mathbb{N}$, ${\alpha}_{2k-1}\neq0,\;{\beta}_{2k-1}=\frac{1}{{\alpha}_{2k-1}},\;{\beta}_{2k}=\frac{\alpha_{2k}}{{\alpha}_{2k-1}\alpha_{2k+1}}$ and $$y_1={\alpha}_1x_1+{\alpha}_2x_2$$ $$y_{2k}={\alpha}_{4k-1}x_{2k}$$ $$y_{2k+1}={\alpha}_{4k}x_{2k}+{\alpha}_{4k+1}x_{2k+1}+{\alpha}_{4k+2}x_{2k+2}$$.
Invertible interpolation on Ax=y in a tridiagonal algebra algl
( Sung Kon Kwak ),( Joo Ho Kang ) 호남수학회 2011 호남수학학술지 Vol.33 No.1
Given vectors x and y in a separable complex Hilbert space H, an interpolating operator is a bounded operator A such that Ax = y. We show the following: let algl be a tridiagonal algebra on a separable complex Hilbert space H and let x=(xi) and y=(yi) be vectors in H.
Kwak, Sung Kon,Kim, Ki Won 대구대학교 기초과학연구소 1999 基礎科學硏究 Vol.15 No.3
In 1965, O. Njastad([1]) introduced the concept of an o-set. Recently S. N. Maheshwari and S. S. Thakur ([2]) defined the notion of α-irresolute mappings. And they showed that the concepts of α-irresolute mappings and continuous mappings are independent. In this paper, we introduce the concept of weakly α-irresolute mappings and investigate some properties of such mappings.
SELF-ADJOINT INTERPOLATION ON Ax = y IN ALG$\cal{L}$
Kwak, Sung-Kon,Kang, Joo-Ho The Korean Society for Computational and Applied M 2011 Journal of applied mathematics & informatics Vol.29 No.3
Given vectors x and y in a Hilbert space $\cal{H}$, an interpolating operator is a bounded operator T such that Tx = y. An interpolating operator for n vectors satisfies the equations $Tx_i=y_i$, for i = 1, 2, ${\cdots}$, n. In this paper the following is proved : Let $\cal{L}$ be a subspace lattice on a Hilbert space $\cal{H}$. Let x and y be vectors in $\cal{H}$ and let $P_x$ be the projection onto sp(x). If $P_xE=EP_x$ for each $E{\in}\cal{L}$, then the following are equivalent. (1) There exists an operator A in Alg$\cal{L}$ such that Ax = y, Af = 0 for all f in $sp(x)^{\perp}$ and $A=A^*$. (2) sup $sup\;\{\frac{{\parallel}E^{\perp}y{\parallel}}{{\parallel}E^{\perp}x{\parallel}}\;:\;E\;{\in}\;{\cal{L}}\}$ < ${\infty}$, $y\;{\in}\;sp(x)$ and < x, y >=< y, x >.