Given operators X and Y acting on a Hilbert space $mathcal{H}$, an interpolating operator is a bounded operator A such that AX = Y. In this article, the following is proved: Let $mathcal{L}$ be a subspace lattice on $mathcal{H}$ and let X and Y be operators acting on a Hilbert space H. Let P be the projection onto the $overline{rangeX}$. If PE = EP for each E ${in}$ $mathcal{L}$, then the following are equivalent: (1) sup ${{frac{{parallel}E^{perp}Yf{parallel}}{{parallel}E^{perp}Xf{parallel}}}:f{in}mathcal{H},;E{in}mathcal{L}}$ < ${infty},;overline{rangeY};{subset};overline{rangeX}$, and there is a bounded operator T acting on $mathcal{H}$ such that < Xf, Tg >=< Yf, Xg >, < Tf, Tg >=< Yf, Yg > for all f and gin $mathcal{H}$ and $T^*h$ = 0 for h ${in};{overline{rangeX}}^{perp}$. (2) There is a normal operator A in AlgL such that AX = Y and Ag = 0 for all g in range ${overline{rangeX}}^{perp}$.