Given two collections of \(n\)-vectors \(\{x_ 1,x_ 2,\dots, x_ n\}\) and \(\{y_ 1, y_ 2,\dots, y_ n\}\) in a Hilbert space \(\mathcal H\) we are interested in the existence of an operator \(T\) such that \(Tx_ i= y_ i\), \(i= 1,\dots, n\). Such a problem is called the \(n\)-vector interpolation problem. To make the problem more interesting we require that \(T\) lie in a fixed algebra \(\mathcal A\), or in some ideal contained in \(\mathcal A\). \textit{R. Kadison} [Proc. Nat. Acad. Sci. U.S.A. 43, 273-276 (1957; Zbl 0078.115)] has considered the problem for \(C^*\)-algebras. \textit{E. C. Lance} [Proc. Lond. Math. Soc., III. Ser. 19, 45-68 (1969; Zbl 0169.175)] has solved the one-vector problem for a nest algebra. His result is extended by \textit{A. Hopenwasser} [Ill. J. Math. 33, No. 4, 657-672 (1989; Zbl 0721.47035)] to the CSL algebras. In the present article necessary and sufficient conditions are given so that the operator \(T\) in the interpolation problem lies in the ideal of Hilbert-Schmidt operators in a CSL algebra.