Some log and weak majorization inequalities in Euclidean Jordan algebras

Abstract

Motivated by Horn's log-majorization (singular value) inequality $s\left(AB\right)\underset{log}{\prec }s\left(A\right)\ast s\left(B\right)$ and the related weak-majorization inequality $s\left(AB\right)\underset{w}{\prec }s\left(A\right)\ast s\left(B\right)$ for square complex matrices, we consider their Hermitian analogs $\lambda \left(\sqrt{A}B\sqrt{A}\right)\underset{log}{\prec }\lambda \left(A\right)\ast \lambda \left(B\right)$ for positive semidefinite matrices and $\lambda (|A\circ B|)\underset{w}{\prec }\lambda \left(|A|\right)\ast \lambda \left(|B|\right)$ for general (Hermitian) matrices, where $A\circ B$ denotes the Jordan product of A and B and $\ast$ denotes the componentwise product in ${\mathcal{R}}^n$. In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form $\lambda (P_{\sqrt{a}}\left(b\right))\underset{log}{\prec }\lambda \left(a\right)\ast \lambda \left(b\right)$ for $a,b\ge 0$ and $\lambda (|a\circ b|)\underset{w}{\prec }\lambda \left(|a|\right)\ast \lambda \left(|b|\right)$ for all a and b, where $P_u$ and $\lambda \left(u\right)$ denote, respectively, the quadratic representation and the eigenvalue vector of an element u. We also describe inequalities of the form $\lambda (|A\bullet b|)\underset{w}{\prec }\lambda \left(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\right(A\left)\right)\ast \lambda \left(|b|\right)$, 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 $\parallel a\circ b{\parallel }_p\le \parallel a{\parallel }_r\parallel b{\parallel }_s$, where $\parallel x{\parallel }_p:=\parallel \lambda \left(x\right){\parallel }_p$ denotes the spectral p-norm of x and $p,q,r\in [1,\mathrm{\infty }]$ 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.

Keywords:

• Euclidean Jordan algebra
• log and weak majorization
• Schur product

AMS Subject Classifications:

• 15A42
• 17C20

Acknowledgments

The second author was financially supported by the National Research Foundation of Korea NRF-2016R1A5A1008055.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The second author was financially supported by the National Research Foundation of Korea NRF-2016R1A5A1008055.