Inequalities for unitarily invariant norms and bilinear matrix products

We give several criteria that are equivalent to the basic singular value majorization inequality (1.1) that is common to both the usual and Hadamard products. We then use these criteria to give a unified proof of the basic majorization inequality for both products. Finally, we introduce natural generalizations of the usual and Hadamard products and show that although these generalizations do not satisfy the majorization inequality, they do satisfy an important weaker inequality that plays a role in establishing their submultiplicativity with respect to every unitarily invariant norm.