INVERTIBLE INTERPOLATION ON Ax = y IN A TRIDIAGONAL ALGEBRA ALGℒ

Abstract

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