In this paper the following is proved: Let $mathcal{L}$ be a subspace lattice on a Hilbert space $mathcal{H}$ and X and Y be operators acting on $mathcal{H}$. Then there exists a compact operator A in $Algmathcal{L}$ such that AX = Y if and only if ${sup}{frac{{parallel}E^{perp}Yf{parallel}}{{parallel}E^{perp}Xf{parallel}};:;f{in}mathcal{H},;E{in}mathcal{L}}$ = K < ${infty}$ and Y is compact. Moreover, if the necessary condition holds, then we may choose an operator A such that AX = Y and ${parallel}A{parallel}=K$.