Some log and weak majorization inequalities in Euclidean Jordan algebras

Abstract

Motivated by Horn's log-majorization (singular value) inequality s(AB) <(log) s(A) * s(B) and the related weak-majorization inequality s(AB) <(w) s(A) * s(B) for square complex matrices, we consider their Hermitian analogs lambda(root AB root A) <(log) lambda(A) * lambda(B) for positive semidefinite matrices and lambda(|A circle B|) <(w) lambda(|A|) * lambda(|B|) for general (Hermitian) matrices, where A. B denotes the Jordan product of A and B and * denotes the componentwise product in R-n. In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form lambda(P-root a(b)) <(log) lambda(a) * lambda(b) for a, b >= 0 and lambda(|a circle b|) <(w) lambda(|a|) * lambda(|b|) for all a and b, where P-u and lambda(u) denote, respectively, the quadratic representation and the eigenvalue vector of an element u. We also describe inequalities of the form lambda(|A . b|) <(w) lambda(diag(A)) * lambda(|b|), where A is a real symmetric positive semidefinite matrix and A . b is the Schur product of A and b. In the form of an application, we prove the generalized Holder type inequality parallel to a circle b parallel to(p) <= parallel to a parallel to(r) parallel to b parallel to(s), where parallel to x parallel to(p) := parallel to lambda(x)parallel to(p) denotes the spectral p-norm of x and p, q, r is an element of [1,8] with 1/p = 1/r + 1/s. We also give precise values of the norms of the Lyapunov transformation La and Pa relative to two spectral p-norms.