Some log and weak majorization inequalities in Euclidean Jordan algebras

Abstract

Motivated by Horn's log-majorization (singular value) inequality s(AB)≺logs(A)∗s(B) and the related weak-majorization inequality s(AB)≺ws(A)∗s(B) for square complex matrices, we consider their Hermitian analogs λ(√AB√A)≺logλ(A)∗λ(B) for positive semidefinite matrices and λ(|A∘B|)≺wλ(|A|)∗λ(|B|) for general (Hermitian) matrices, where A∘B denotes the Jordan product of A and B and ∗ denotes the componentwise product in Rⁿ. In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form λ(P√ₐ(b))≺logλ(a)∗λ(b) for a,b≥0 and λ(|a∘b|)≺wλ(|a|)∗λ(|b|) for all a and b, where Pᵤ and λ(u) denote, respectively, the quadratic representation and the eigenvalue vector of an element u. We also describe inequalities of the form λ(|A∙b|)≺wλ(diag(A))∗λ(|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 Hölder type inequality ||a∘b||p≤||a||r||b||s, where ||x||p:=||λ(x)||p denotes the spectral p-norm of x and p,q,r∈[1,∞] 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.