INVERTIBLE INTERPOLATION ON Ax = y IN A TRIDIAGONAL ALGEBRA ALGℒ

• Journal title : Honam Mathematical Journal
• Volume 33, Issue 1,  2011, pp.115-120
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2011.33.1.115
Title & Authors
INVERTIBLE INTERPOLATION ON Ax = y IN A TRIDIAGONAL ALGEBRA ALGℒ
Kwak, Sung-Kon; Kang, Joo-Ho;

Abstract
Given vectors x and y in a separable complex Hilbert space $\small{\cal{H}}$, an interpolating operator is a bounded operator A such that Ax = y. We show the following : Let Alg$\small{\cal{L}}$ be a tridiagonal algebra on a separable complex Hilbert space H and let x = ($\small{x_i}$) and y = ($\small{y_i}$) be vectors in H. Then the following are equivalent: (1) There exists an invertible operator A = ($\small{a_{kj}}$) in Alg$\small{\cal{L}}$ such that Ax = y. (2) There exist bounded sequences $\small{\{{\alpha}_n\}}$ and $\small{\{{{\beta}}_n\}}$ in $\small{\mathbb{C}}$ such that for all $\small{k\in\mathbb{N}}$, $\small{{\alpha}_{2k-1}\neq0,\;{\beta}_{2k-1}=\frac{1}{{\alpha}_{2k-1}},\;{\beta}_{2k}=\frac{\alpha_{2k}}{{\alpha}_{2k-1}\alpha_{2k+1}}}$ and $\small{y_1={\alpha}_1x_1+{\alpha}_2x_2}$ $\small{y_{2k}={\alpha}_{4k-1}x_{2k}}$ $\small{y_{2k+1}={\alpha}_{4k}x_{2k}+{\alpha}_{4k+1}x_{2k+1}+{\alpha}_{4k+2}x_{2k+2}}$.
Keywords
Invertible Interpolation;CSL-Algebra;Tridiagonal Al-gebra;Alg$\small{\cal{L}}$;
Language
English
Cited by
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