Some log and weak majorization inequalities in Euclidean Jordan algebras

Motivated by Horn’s log-majorization (singular value) inequality $s(AB)s(A)\ast s(B)$ and the related weak-majorization inequality $s(AB)s(A)\ast s(B)$ for square complex matrices, we consider their Hermitian analogs $\lambda(\sqrt{A}B\sqrt{A}) \underset{log}{\prec} \lambda(A)\lambda(B)$ for positive semidefinite matrices and $\lambda (\mathrm{\mid }A\circ B\mathrm{\mid })\lambda (\mathrm{\mid }A\mathrm{\mid })\ast \lambda (\mathrm{\mid }B\mathrm{\mid })$ 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 $\lambda\big (P_{\sqrt{a}}(b)\big )\underset{log}{\prec} \lambda(a)\lambda(b)$for$a,b\geq 0$and$\lambda\big (|a\circ b|\big )\underset{w}{\prec} \lambda(|a|)\lambda(|b|)$forall$a$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{d}\mathrm{i}\mathrm{a}\mathrm{g}$}(A))\lambda(|b|)$,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 $\mathrm{\mid }\mathrm{\mid }a\circ b\mathrm{\mid }{\mathrm{\mid }}_p\le \mathrm{\mid }\mathrm{\mid }a\mathrm{\mid }{\mathrm{\mid }}_r\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }{\mathrm{\mid }}_s$, where $\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }{\mathrm{\mid }}_p:=\mathrm{\mid }\mathrm{\mid }\lambda (x)\mathrm{\mid }{\mathrm{\mid }}_p$ 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.