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 paper, we showed the following : Let $mathcal{L}$ be a subspace lattice acting on a Hilbert space $mathcal{H}$ and let $X_i$ and $Y_i$ be operators in B($mathcal{H}$) for i = 1, 2, ${cdots}$. Let $P_i$ be the projection onto $overline{rangeX_i}$ for all i = 1, 2, ${cdots}$. If $P_kE$ = $EP_k$ for some k in $mathbb{N}$ and all E in $mathcal{L}$, then the following are equivalent: (1) $sup;{{frac{{parallel}E^{perp}({sum}^n_{i=1}Y_if_i){parallel}}{{parallel}E^{perp}({sum}^n_{i=1}Y_if_i){parallel}}:f{in}H,n{in}{mathbb{N}},E{in}mathcal{L}}}$ < ${infty}$ range $overline{rangeY_k};=;overline{rangeX_k};=;mathcal{H}$, and < $X_kf,;X_kg$ >=< $Y_kf,;Y_kg$ > for some k in $mathbb{N}$ and for all f and g in $mathcal{H}$. (2) There exists an operator A in Alg$mathcal{L}$ such that $AX_i$ = $Y_i$ for i = 1, 2, ${cdots}$ and AA$^*$ = I = A$^*$A.