On singular values of products of matrices and log-majorization

In this paper, we are concerned with the problem of improving the classical inequality s (XY) (sic)(log) s(X) circle s(Y), where X, Y are n x n matrices, s(A) denotes the vector of singular values of a matrix A, circle is the Schur product on R-n and (sic)(log) stands for the log-majorization preorder on R-n. We show that s (XY) (sic)(log) s(XZ) circle s(W) (sic)(log)s(X) circle s(Y)for some special matrices Z and W depending on Y. Moreover, we prove that the operator Z -> s(XZ) circle s(Z)([-1]) circle s(Y) is monotone. To this end we introduce an adequate preorder on the matrix space M-n. Some related results are also demonstrated.