Some log and weak majorization inequalities in Euclidean Jordan algebras

Jiyuan Tao, Juyoung Jeong, Muddappa Gowda

Published 2020-03-27Version 1

In the setting of Euclidean Jordan algebras, we prove majorization inequalities $\lambda\big (P_{\sqrt{a}}(b)\big )\underset{log}{\prec} \lambda(a)*\lambda(b)$ when $a,b\geq 0$, $\lambda\big (|P_{a}(b)|\big )\underset{w}{\prec} \lambda(a^2)*\lambda(|b|)$ 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 $u$, and $*$ denotes the componentwise product. Extending the second inequality, we show that $\lambda(|A\bullet b|)\underset{w}{\prec} \lambda({\rm{diag}}(A))*\lambda(|b|)$, 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 applications, we prove the generalized H\"{o}lder type inequality $||a\circ b||_p\leq ||a||_r\,||b||_s$, where $p,q,r\in [1,\infty]$ with $\frac{1}{p}=\frac{1}{r}+\frac{1}{s}$ and compute the norms of Lyapunov transformation $L_a$ and quadratic representation $P_a$ relative to spectral norms.