Given operators X and Y acting on a Hilbert space $cal{H}$, an interpolating operator is a bounded operator A such that AX = Y. In this article, we showed the following : Let $cal{L}$ be a subspace lattice acting on a Hilbert space $cal{H}$ and let X and Y be operators in $cal{B}(cal{H})$. Let P be the projection onto $ar{rangeX}$. If FE = EF for every $Eincal{L}$, then the following are equivalent: (1) $sup{{{parallel}E^{perp}Yfparallelatop parallel{E}^{perp}Xfparallel};:;f{in}cal{H},;Eincal{L}}$ < $infty$, $ar{range;Y}subsetar{range;X}$, and < Xf, Yg >=< Yf,Xg > for any f and g in $cal{H}$. (2) There exists a self-adjoint operator A in Alg$cal{L}$ such that AX = Y.