• Title, Summary, Keyword: tridiagonal algebra

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COMPACT INTERPOLATION FOR VECTORS IN TRIDIAGONAL ALGEBRA

  • Jo, Young-Soo;Kang, Joo-Ho
    • Communications of the Korean Mathematical Society
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    • v.18 no.3
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    • pp.485-490
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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,…,n. In this article, we investigate compact interpolation problems in tridiagonal algebra : Given vectors x and y in a Hilbert space, when is there a compact operator A in a tridiagonal algebra such that Ax = y?

SELF-ADJOINT INTERPOLATION ON Ax = Y IN A TRIDIAGONAL ALGEBRA ALGL

  • PARK, DONGWAN;PARK, JAE HYUN
    • Honam Mathematical Journal
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    • v.28 no.1
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    • pp.135-140
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    • 2006
  • Given vectors x and y in a separable Hilbert space H, an interpolating operator is a bounded operator A such that Ax = y. In this article, we investigate self-adjoint interpolation problems for vectors in a tridiagonal algebra: Let AlgL 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 a self-adjoint operator $A=(a_ij)$ in AlgL such that Ax = y. (2) There is a bounded real sequence {$a_n$} such that $y_i=a_ix_i$ for $i{\in}N$.

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COMPACT INTERPOLATION ON Ax = y IN A TRIDIAGONAL ALGEBRA ALG$\mathcal{L}$

  • Kang, Joo-Ho
    • Honam Mathematical Journal
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    • v.32 no.2
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    • pp.255-260
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    • 2010
  • Given vectors x and y in a separable complex Hilbert space $\mathcal{H}$, 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. We show the following : Let Alg$\mathcal{L}$ be a tridiagonal algebra on a separable complex Hilbert space $\mathcal{H}$ and let x = $(x_i)$ and y = $(y_i)$ be vectors in H. Then the following are equivalent: (1) There exists a compact operator A = $(a_{ij})$ in Alg$\mathcal{L}$ such that Ax = y. (2) There is a sequence ${{\alpha}_n}$ in $\mathbb{C}$ such that ${{\alpha}_n}$ converges to zero and for all k ${\in}$ $\mathbb{N}$, $y_1 = {\alpha}_1x_1 + {\alpha}_2x_2$ $y_{2k} = {\alpha}_{4k-1}x_{2k}$ $y_{2k+1}={\alpha}_{4k}x_{2k}+{\alpha}_{4k+1}x_{2k+1}+{\alpha}_{4k+2}+x_{2k+2}$.

SELF-ADJOINT INTERPOLATION ON AX=Y IN A TRIDIAGONAL ALGEBRA ALG𝓛

  • Kang, Joo Ho;Lee, SangKi
    • Honam Mathematical Journal
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    • v.36 no.1
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    • pp.29-32
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    • 2014
  • Given operators X and Y acting on a separable Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that AX = Y. In this article, we investigate self-adjoint interpolation problems for operators in a tridiagonal algebra : Let $\mathcal{L}$ be a subspace lattice acting on a separable complex Hilbert space $\mathcal{H}$ and let X = ($x_{ij}$) and Y = ($y_{ij}$) be operators acting on $\mathcal{H}$. Then the following are equivalent: (1) There exists a self-adjoint operator A = ($a_{ij}$) in $Alg{\mathcal{L}}$ such that AX = Y. (2) There is a bounded real sequence {${\alpha}_n$} such that $y_{ij}={\alpha}_ix_{ij}$ for $i,j{\in}\mathbb{N}$.

UNITARY INTERPOLATION ON Ax = y IN A TRIDIAGONAL ALGEBRA ALG𝓛

  • Kang, Joo Ho
    • Honam Mathematical Journal
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    • v.36 no.4
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    • pp.907-911
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    • 2014
  • Given vectors x and y in a separable complex Hilbert space $\mathcal{H}$, an interpolating operator is a bounded operator A such that Ax = y. We show the following: Let $Alg{\mathcal{L}}$ be a tridiagonal algebra on $\mathcal{H}$ and let $x=(x_i)$ and $y=(y_i)$ be vectors in $\mathcal{H}$. Then the following are equivalent: (1) There exists a unitary operator $A=(a_{ij})$ in $Alg{\mathcal{L}}$ such that Ax = y. (2) There is a bounded sequence $\{{\alpha}_i\}$ in $\mathbb{C}$ such that ${\mid}{\alpha}_i{\mid}=1$ and $y_i={\alpha}_ix_i$ for $i{\in}\mathbb{N}$.

ISOMETRIES OF (2)}_{2n}$

  • Park, Taeg-Young
    • Communications of the Korean Mathematical Society
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    • v.10 no.3
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    • pp.609-620
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    • 1995
  • In this paper, we introduce the generalization $A^{(2)}_{2n}$ of tridiagonal algebras $A_{2n}$ and investigate the isometries of such algebras.

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HILBERT-SCHMIDT INTERPOLATION ON AX=Y IN A TRIDIAGONAL ALGEBRA ALG${\pounds}$

  • Kang, Joo-Ho
    • The Pure and Applied Mathematics
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    • v.15 no.4
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    • pp.401-406
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    • 2008
  • Given operators X and Y acting on a separable complex Hilbert space H, an interpolating operator is a bounded operator A such that AX=Y. In this article, we investigate Hilbert-Schmidt interpolation problems for operators in a tridiagonal algebra and we get the following: Let ${\pounds}$ be a subspace lattice acting on a separable complex Hilbert space H and let X=$(x_{ij})$ and Y=$(y_{ij})$ be operators acting on H. Then the following are equivalent: (1) There exists a Hilbert-Schmidt operator $A=(a_{ij})$ in Alg${\pounds}$ such that AX=Y. (2) There is a bounded sequence $\{{\alpha}_n\}$ in $\mathbb{C}$ such that ${\sum}_{n=1}^{\infty}|{\alpha}_n|^2<{\infty}$ and $$y1_i={\alpha}_1x_{1i}+{\alpha}_2x_{2i}$$ $$y2k_i={\alpha}_{4k-1}x_2k_i$$ $$y{2k+1}_i={\alpha}_{4k}x_{2k}_i+{\alpha}_{4k+1}x_{2k+1}_i+{\alpha}_{4k+2}x_{2k+2}_i\;for\;all\;i,\;k\;\mathbb{N}$$.

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HILBERT-SCHMIDT INTERPOLATION ON Ax = y IN A TRIDIAGONAL ALGEBRA ALGL

  • Jo, Young-Soo;Kang, Joo-Ho
    • The Pure and Applied Mathematics
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    • v.11 no.2
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    • pp.167-173
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    • 2004
  • Given vectors x and y in a separable Hilbert space $\cal H$, an interpolating operator is a bounded operator A such that Ax = y. In this article, we investigate Hilbert-Schmidt interpolation problems for vectors in a tridiagonal algebra. We show the following: Let $\cal L$ be a subspace lattice acting on a separable complex Hilbert space $\cal H$ and let x = ($x_{i}$) and y = ($y_{i}$) be vectors in $\cal H$. Then the following are equivalent; (1) There exists a Hilbert-Schmidt operator A = ($a_{ij}$ in Alg$\cal L$ such that Ax = y. (2) There is a bounded sequence {$a_n$ in C such that ${\sum^{\infty}}_{n=1}\mid\alpha_n\mid^2 < \infty$ and $y_1 = \alpha_1x_1 + \alpha_2x_2$ ... $y_{2k} =\alpha_{4k-1}x_{2k}$ $y_{2k=1} = \alpha_{4kx2k} + \alpha_{4k+1}x_{2k+1} + \alpha_{4k+1}x_{2k+2}$ for K $\epsilon$ N.

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ISOMORPHISMS OF CERTAIN TRIDIAGONAL ALGEBRAS

  • Choi, Taeg-Young;Kim, Si-Ju
    • The Pure and Applied Mathematics
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    • v.7 no.1
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    • pp.49-60
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    • 2000
  • We will characterize isomorphisms from the adjoint of a certain tridiag-onal algebra $AlgL_{2n}$ onto $AlgL_{2n}$. In this paper the following are proved: A map $\Phi{\;}:{\;}(AlgL_{2n})^{*}{\;}{\longrightarrow}{\;}AlgL_{2n}$ is an isomorphism if and only if there exists an operator S in $AlgL_{2n}$ with all diagonal entries are 1 and an invertible backward diagonal operator B such that ${\Phi}(A){\;}={\;}SBAB^{-1}S^{-1}$.

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