Given vectors x and y in a Hilbert space $cal{H}$, an interpolating operator is a bounded operator T such that Tx = y. An interpolating operator for n vectors satisfies the equations $Tx_i=y_i$, for i = 1, 2, ${cdots}$, n. In this paper the following is proved : Let $cal{L}$ be a subspace lattice on a Hilbert space $cal{H}$. Let x and y be vectors in $cal{H}$ and let $P_x$ be the projection onto sp(x). If $P_xE=EP_x$ for each $E{in}cal{L}$, then the following are equivalent. (1) There exists an operator A in Alg$cal{L}$ such that Ax = y, Af = 0 for all f in $sp(x)^{perp}$ and $A=A^*$. (2) sup $sup;{frac{{parallel}E^{perp}y{parallel}}{{parallel}E^{perp}x{parallel}};:;E;{in};{cal{L}}}$ < ${infty}$, $y;{in};sp(x)$ and < x, y >=< y, x >.