Some log and weak majorization inequalities in Euclidean Jordan algebras

Abstract

Motivated by Horn's log-majorization (singular value) inequality $s(AB)\underset{log}{\prec} s(A)*s(B)$ and the related weak-majorization inequality $s(AB)\underset{w}{\prec} s(A)*s(B)$ for square complex matrices, we consider their Hermitian analogs $\lambda(\sqrt{A}B\sqrt{A}) \underset{log}{\prec} \lambda(A)*\lambda(B)$ for positive semidefinite matrices and $\lambda(|A\circ B|) \underset{w}{\prec} \lambda(|A|)*\lambda(|B|)$ for general (Hermitian) matrices, where $A\circ B$ 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 $\lambda\big (P_{\sqrt{a}}(b)\big )\underset{log}{\prec} \lambda(a)*\lambda(b)$ for $a,b\geq 0$ and $\lambda\big (|a\circ b|\big )\underset{w}{\prec} \lambda(|a|)*\lambda(|b|)$ 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 $\lambda(|A\bullet b|)\underset{w}{\prec} \lambda({\mathrm{diag}}(A))*\lambda(|b|)$, where $A$ is a real symmetric positive semidefinite matrix and $A\,\bullet\, b$ is the Schur product of $A$ and $b$. In the form of an application, we prove the generalized H\"{o}lder type inequality $||a\circ b||_p\leq ||a||_r\,||b||_s$, where $||x||_p:=||\lambda(x)||_p$ denotes the spectral $p$-norm of $x$ and $p,q,r\in [1,\infty]$ with $\frac{1}{p}=\frac{1}{r}+\frac{1}{s}$. We also give precise values of the norms of the Lyapunov transformation $L_a$ and $P_a$ relative to two spectral $p$-norms.