Given vectors x and y in a Hilbert space, an interpolating operator is a bounded operator T such that Tx = y. An interpolating operator for n vectors satisfies the equation $Tx_i;=;y_i,;for;i;=;1,;2,;cdots,;n$. In this paper the following is proved: Let H be a Hilbert space and L be a commutative subspace lattice on H. Let H and y be vectors in H. Let $M_x;=;{{sum{n}{i=1}};{alpha}_iE_ix;:;n;{in};N,;{alpha}_i;{in};{mathbb{C}};and;E_i;{in};L};and;M_y;=;{{sum{n}{i=1}};{alpha}_iE_iy;:;n;{in};N,;{alpha}_i;{in};{mathbb{C}};and;E_i;{in};L}. Then the following are equivalent. (1) There exists an operator A in AlgL such that Ax = y, Af = 0 for all f in ${overline{M_x}}^{ot}$, AE = EA for all $E;{in};L;and;A^{*};=;A$. (2) $sup;{frac{{parallel}{{Sigma}_{i=1}}^{n};{alpha}_iE_iy{parallel}}{{parallel}{{Sigma}_{i=1}}^{n};{alpha}_iE_iy{parallel}};:;n;{in};N,;{alpha}_i;{in};{mathbb{C}};and;E_i;{in};L};<;{infty},;{overline{M_u}};{subset}{overline{M_x}}$ and < Ex, y >=< Ey, x > for all E in L.