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