On operator interpolation problems

In this paper we obtained the following : Let H be a Hilbert space and L be a subspace lattice on H. Let X and Y be operators acting on X If the range of X is dense in H, then the following are equivalent: (1) there exists an operator A in AlgL such that AX = Y, (2) sup {parallel toE(perpendicular to)Yfparallel to/parallel toE(perpendicular to)Xfparallel to : f is an element of H, E is an element of L } = K < infinity. Moreover, if condition (2) holds, we may choose the operator A such that parallel toAparallel to = K.