Some log and weak majorization inequalities in Euclidean Jordan algebras

Abstract

Motivated by Horn's log-majorization (singular value) inequality $s(AB)\underset{log}{\prec }s(A)\ast s(B)$ and the related weak-majorization inequality $s(AB)\underset{w}{\prec }s(A)\ast s(B)$ for square complex matrices, we consider their Hermitian analogs $\lambda (\sqrt{A}B\sqrt{A})\underset{log}{\prec }\lambda (A)\ast \lambda (B)$ for positive semidefinite matrices and $\lambda (|A\circ B|)\underset{w}{\prec }\lambda (|A|)\ast \lambda (|B|)$ for general (Hermitian) matrices, where $A\circ B$ denotes the Jordan product of $A$ and $B$ and $\ast$ denotes the componentwise product in $R^n$. In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form $\lambda (P_{\sqrt{a}}(b))\underset{log}{\prec }\lambda (a)\ast \lambda (b)$ for $a,b\ge 0$ and $\lambda (|a\circ b|)\underset{w}{\prec }\lambda (|a|)\ast \lambda (|b|)$ 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 $\lambda (|A\bullet b|)\underset{w}{\prec }\lambda (\mathrm{diag}(A))\ast \lambda (|b|)$, 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 Hölder type inequality $||a\circ b|{|}_p\le ||a|{|}_r||b|{|}_s$, 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.

18 pages; abstract, introduction, and final section rewritten, references updated