Some log and weak majorization inequalities in Euclidean Jordan algebras

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 $λ(\sqrt{A}B\sqrt{A}) \underset{log}{\prec} λ(A)*λ(B)$ for positive semidefinite matrices and $λ(|A\circ B|) \underset{w}{\prec} λ(|A|)*λ(|B|)$ 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 $λ\big (P_{\sqrt{a}}(b)\big )\underset{log}{\prec} λ(a)*λ(b)$ for $a,b\geq 0$ and $λ\big (|a\circ b|\big )\underset{w}{\prec} λ(|a|)*λ(|b|)$ for all $a$ and $b$, where $P_u$ and $λ(u)$ denote, respectively, the quadratic representation and the eigenvalue vector of an element $u$. We also describe inequalities of the form $λ(|A\bullet b|)\underset{w}{\prec} λ({\mathrm{diag}}(A))*λ(|b|)$, 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 $||a\circ b||_p\leq ||a||_r\,||b||_s$, where $||x||_p:=||λ(x)||_p$ 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.

No reviews yet.

No comments yet.