Abstract

Given operators X and Y acting on a separable complex Hilbert space H, an interpolating operator is a bounded operator A such that AX=Y. In this article, we investigate Hilbert-Schmidt interpolation problems for operators in a tridiagonal algebra and we get the following: Let <TEX>${\pounds}$</TEX> be a subspace lattice acting on a separable complex Hilbert space H and let X=<TEX>$(x_{ij})$</TEX> and Y=<TEX>$(y_{ij})$</TEX> be operators acting on H. Then the following are equivalent: (1) There exists a Hilbert-Schmidt operator <TEX>$A=(a_{ij})$</TEX> in Alg<TEX>${\pounds}$</TEX> such that AX=Y. (2) There is a bounded sequence <TEX>$\{{\alpha}_n\}$</TEX> in <TEX>$\mathbb{C}$</TEX> such that <TEX>${\sum}_{n=1}^{\infty}|{\alpha}_n|^2</TEX><TEX><</TEX><TEX>{\infty}$</TEX> and <TEX> $$y1_i={\alpha}_1x_{1i}+{\alpha}_2x_{2i}$$ </TEX> <TEX> $$y2k_i={\alpha}_{4k-1}x_2k_i$$ </TEX> <TEX> $$y{2k+1}_i={\alpha}_{4k}x_{2k}_i+{\alpha}_{4k+1}x_{2k+1}_i+{\alpha}_{4k+2}x_{2k+2}_i\;for\;all\;i,\;k\;\mathbb{N}$$ </TEX>.