On unitarily invariant norms of matrix-valued linear positive operators

In this paper we prove several inequalities concerning invariant norms of matrices belonging to the range of some matrix-valued Linear Positive Operator (LPO). We provide a variational characterization of unitarily invariant norms in terms of bilinear forms and a kind of Cauchy-Schwarz inequality for matrix-valued LPOs. The latter inequality holds for matrix-valued LPOs acting on Lp spaces (e.g., multi-level Toeplitz, Finite Elements matrices etc.) but it is still unclear if it is true in general. These tools turn out to be very effective in order to deduce inequalities concerning norms of multilevel Toeplitz matrices and of some related approximations in matrix algebras.