NORMAL INTERPOLATION ON AX = Y IN ALG<TEX>$\mathcal{L}$</TEX>

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 =, = 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 .