Some inequalities for Hadamard products of matrices

Let C(n) and R(n)(+) denote the sets of n x n complex and nonnegative real matrices respectively, and parallel to center dot parallel to(F) be the Frobenius norm. We first characterize those unitarily invariant norms parallel to center dot parallel to satisfying parallel to[a(ij)]parallel to = parallel to[vertical bar a(ij)vertical bar]parallel to for all [a(ij)] is an element of C(n). We then prove that if parallel to center dot parallel to is a norm on C(n), then vertical bar vertical bar A circle X circle B parallel to(2) <= parallel to A circle X circle(A) over bar parallel to parallel to B circle X circle(B) over bar parallel to for all X is an element of R(n)(+), A, B is an element of C(n) if and only if the norm parallel to center dot parallel to satisfies parallel to[C(ij)]parallel to <= parallel to[vertical bar C(ij)vertical bar]parallel to for all [C(ij)] is an element of C(n), and that if A, B, X is an element of C(n), then A circle X circle B vertical bar(p)parallel to(2) <= n parallel to vertical bar A circle X circle(A) over bar vertical bar(p)parallel to parallel to vertical bar B circle X circle(B) over bar vertical bar(p)parallel to (p >= 1) for any unitarily invariant norm parallel to center dot parallel to, where vertical bar A vertical bar = (A*A)(1/2). These results are related to some work of R. A. Horn and R. Mathias and work of R. Bhatia, C. Davis and M. D. Choi.