• Bulletin of the Korean Mathematical Society
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• v.40 no.2
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• pp.207-213
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• 2003

Given vectors x and y in a Hilbert space, an interpolating operator is a bounded operator T such that Tx=y. An interpolating operator for n-vectors satisfies the equation Ax$_{i}$=y$_{i}$. for i=1,2, …, n. In this article, we investigate unitary interpolation problems in CSL-Algebra AlgL : Let L be a commutative subspace lattice on a Hilbert space H. Let x and y be vectors in H. When does there exist a unitary operator A in AlgL such that Ax=y?

• Journal of applied mathematics & informatics
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• v.12 no.1_2
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• pp.359-365
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• 2003

Given vectors x and y in a Hilbert space, an interpolating operator is a bounded operator T such that Tx = y. In this article, we investigate invertible interpolation problems in CSL-Algebra AlgL : Let L be a commutative subspace lattice on a Hilbert space H and x and y be vectors in H. When does there exist an invertible operator A in AlgL suth that An = ㅛ?

• Communications of the Korean Mathematical Society
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• v.13 no.1
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• pp.7-20
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• 1998

In this paper, we will investigated certain two types of isomorphisms of $B^{(n)}_{2n}$ which are closely related.

• Bulletin of the Korean Mathematical Society
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• v.33 no.2
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• pp.233-241
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• 1996

The study of non-self-adjoint operator algebras on Hilbert space was only beginned by W.B. Arveson[1] in 1974. Recently, such algebras have been found to be of use in physics, in electrical engineering, and in general systems theory. Of particular interest to mathematicians are reflexive algebras with commutative lattices of invariant subspaces.

• Kyungpook Mathematical Journal
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• v.45 no.2
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• pp.293-299
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• 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
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• v.15 no.1_2
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• pp.503-510
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• 2004

Given vectors x and y in a Hilbert space, an interpolating operator is a bounded operator T such that Tx = y. An interpolating operator for n vectors satisfies the equation $Tx_i\;=\;y_i,\;for\;i\;=\;1,\;2,\;\cdots,\;n$. In this paper the following is proved: Let H be a Hilbert space and L be a commutative subspace lattice on H. Let H and y be vectors in H. Let $M_x\;=\;\{{\sum{n}{i=1}}\;{\alpha}_iE_ix\;:\;n\;{\in}\;N,\;{\alpha}_i\;{\in}\;{\mathbb{C}}\;and\;E_i\;{\in}\;L\}\;and\;M_y\;=\;\{{\sum{n}{i=1}}\;{\alpha}_iE_iy\;:\;n\;{\in}\;N,\;{\alpha}_i\;{\in}\;{\mathbb{C}}\;and\;E_i\;{\in}\;L\}. Then the following are equivalent. (1) There exists an operator A in AlgL such that Ax = y, Af = 0 for all f in ${\overline{M_x}}^{\bot}$, AE = EA for all $E\;{\in}\;L\;and\;A^{*}\;=\;A$. (2) $sup\;\{\frac{{\parallel}{{\Sigma}_{i=1}}^{n}\;{\alpha}_iE_iy{\parallel}}{{\parallel}{{\Sigma}_{i=1}}^{n}\;{\alpha}_iE_iy{\parallel}}\;:\;n\;{\in}\;N,\;{\alpha}_i\;{\in}\;{\mathbb{C}}\;and\;E_i\;{\in}\;L\}\;<\;{\infty},\;{\overline{M_u}}\;{\subset}{\overline{M_x}}$ and < Ex, y >=< Ey, x > for all E in L.

• Honam Mathematical Journal
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• v.26 no.4
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• pp.379-389
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• 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 $f_i{\in}H$.

• Journal of applied mathematics & informatics
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• v.11 no.1_2
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• pp.431-436
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• 2003

Given vectors x and y in a Hilbert space, an interpolating operator is a bounded operator T such that Tx = y. An interpolating operator for n vectors satisfies the equation $Tx_i\;:\;y_i,\;for\;i\;=\;1,\;2,\;{\cdots},\;n$. In this article, we obtained the following : $Let\;x\;=\;\{x_i\}\;and\;y=\{y_\}$ be two vectors in a separable complex Hilbert space H such that $x_i\;\neq\;0$ for all $i\;=\;1,\;2;\cdots$. Let L be a commutative subspace lattice on H. Then the following statements are equivalent. (1) $sup\;\{\frac{\$\mid${\sum_{k=1}}^l\;\alpha_{\kappa}E_{\kappa}y\$\mid$}{\$\mid${\sum_{k=1}}^l\;\alpha_{\kappa}E_{\kappa}x\$\mid$}\;:\;l\;\in\;\mathbb{N},\;\alpha_{\kappa}\;\in\;\mathbb{C}\;and\;E_{\kappa}\;\in\;L\}\;<\;\infty\;and\;$\mid$y_n\$\mid$x_n$\mid$^{-1}\;=\;1\;for\;all\;n\;=\;1,\;2,\;\cdots$. (2) There exists an operator A in AlgL such that Ax = y, A is a unitary operator and every E in L reduces, A, where AlgL is a tridiagonal algebra.

• Bulletin of the Korean Mathematical Society
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• v.39 no.1
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• pp.63-69
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• 2002

Given vectors x and y in a Hilbert space, an intepolating operator is a bounded operator T such that Tx=y. an interpolating operator for n vectors satisfies the equation Tx$_{i}$=y, for i=1, 2,…, n. In this article, we obtained the fellowing : Let x = (x$_{i}$) and y = (y$_{i}$) be two vectors in H such that x$_{i}$$\neq$0 for all i = 1, 2,…. Then the following statements are equivalent. (1) There exists an operator A in AlgＬ such that Ax = y, A is a trace-class operator and every E in Ｌ reduces A. (2) (equation omitted).mitted).