Solving operator equations in nest algebras

Let X and Y be operators on Hilbert space, and let L be a nest of projections on the space. We consider the problem of finding an operator A in Alg L such that A is Hilbere-Schmidt and such that AX = Y. A necessary and sufficient condition involving X,Y, and the projections in the lattice is found. We also indicate how the statements of the results can be modified so that the main theorem is true for any commutative subspace lattice L.