Given a nest algebra \(\text{Alg }N\) of operators in a Hilbert space \(H\), the authors characterize the pairs \(X\), \(Y\) of operators on \(H\) for which there exists a compact element \(A\) in \(\text{Alg }N\) which satisfies the equation \(AX= Y\). They solve the same interpolation problem for a system of equations \(Ax_ i= y_ i\) \((i= 1,\dots, n)\); \(x_ i\), \(y_ i\) fixed vectors in \(H\), and some problems related to the one vector interpolation equation, where the solution is asked to belong to a CSL algebra and satisfy additional conditions.