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INVERTIBLE INTERPOLATION ON Ax = y IN A TRIDIAGONAL ALGEBRA ALGℒ

  • Kwak, Sung-Kon (Department of Mathematics, Daegu University) ;
  • Kang, Joo-Ho (Department of Mathematics, Daegu University)
  • Received : 2011.02.24
  • Accepted : 2011.03.11
  • Published : 2011.03.25

Abstract

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 $$y_1={\alpha}_1x_1+{\alpha}_2x_2$$ $$y_{2k}={\alpha}_{4k-1}x_{2k}$$ $$y_{2k+1}={\alpha}_{4k}x_{2k}+{\alpha}_{4k+1}x_{2k+1}+{\alpha}_{4k+2}x_{2k+2}$$.

Keywords

Invertible Interpolation;CSL-Algebra;Tridiagonal Al-gebra;Alg$\cal{L}$

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