Characterization of singular numbers of products of operators in matrix algebras and finite von Neumann algebras

We characterize in terms of inequalities the possible generalized singular numbers of a product AB of operators A and B   having given generalized singular numbers, in an arbitrary finite von Neumann algebra. We also solve the analogous problem in matrix algebras Mn(C)Mn(C), which seems to be new insofar as we do not require A and B to be invertible.