Given operators X and Y acting on a Hilbert space

, an interpolating operator is a bounded operator A such that AX = Y. In this article, the following is proved: Let

be a subspace lattice on

and let X and Y be operators acting on a Hilbert space H. Let P be the projection onto the

. If PE = EP for each E

, then the following are equivalent: (1) sup

<

, and there is a bounded operator T acting on

such that < Xf, Tg >=< Yf, Xg >, < Tf, Tg >=< Yf, Yg > for all f and gin

and

= 0 for h

. (2) There is a normal operator A in AlgL such that AX = Y and Ag = 0 for all g in range

.