Abstract

Given vectors x and y in a separable Hilbert space <TEX>$\cal H$</TEX>, 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 <TEX>$\cal L$</TEX> be a subspace lattice acting on a separable complex Hilbert space <TEX>$\cal H$</TEX> and let x = (<TEX>$x_{i}$</TEX>) and y = (<TEX>$y_{i}$</TEX>) be vectors in <TEX>$\cal H$</TEX>. Then the following are equivalent; (1) There exists a Hilbert-Schmidt operator A = (<TEX>$a_{ij}$</TEX> in Alg<TEX>$\cal L$</TEX> such that Ax = y. (2) There is a bounded sequence {<TEX>$a_n$</TEX> in C such that <TEX>${\sum^{\infty}}_{n=1}\mid\alpha_n\mid^2 < \infty$</TEX> and <TEX>$y_1 = \alpha_1x_1 + \alpha_2x_2$</TEX> ... <TEX>$y_{2k} =\alpha_{4k-1}x_{2k}$</TEX> <TEX>$y_{2k=1} = \alpha_{4kx2k} + \alpha_{4k+1}x_{2k+1} + \alpha_{4k+1}x_{2k+2}$</TEX> for K <TEX>$\epsilon$</TEX> N.