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

• Journal title : Honam Mathematical Journal
• Volume 32, Issue 2,  2010, pp.255-260
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2010.32.2.255
Title & Authors
COMPACT INTERPOLATION ON Ax = y IN A TRIDIAGONAL ALGEBRA ALG$\small{\mathcal{L}}$
Kang, Joo-Ho;

Abstract
Given vectors x and y in a separable complex Hilbert space $\small{\mathcal{H}}$, 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$\small{\mathcal{L}}$ be a tridiagonal algebra on a separable complex Hilbert space $\small{\mathcal{H}}$ and let x = $\small{(x_i)}$ and y = $\small{(y_i)}$ be vectors in H. Then the following are equivalent: (1) There exists a compact operator A = $\small{(a_{ij})}$ in Alg$\small{\mathcal{L}}$ such that Ax = y. (2) There is a sequence $\small{{{\alpha}_n}}$ in $\small{\mathbb{C}}$ such that $\small{{{\alpha}_n}}$ converges to zero and for all k $\small{{\in}}$ $\small{\mathbb{N}}$, $\small{y_1 = {\alpha}_1x_1 + {\alpha}_2x_2}$ $\small{y_{2k} = {\alpha}_{4k-1}x_{2k}}$ $\small{y_{2k+1}={\alpha}_{4k}x_{2k}+{\alpha}_{4k+1}x_{2k+1}+{\alpha}_{4k+2}+x_{2k+2}}$.
Keywords
Compact Interpolation;CSL-Algebra;Tridiagonal Algebra;Alg$\small{\mathcal{L}}$;
Language
English
Cited by
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