论文标题
欧几里得约旦代数的一些日志和弱化不平等
Some log and weak majorization inequalities in Euclidean Jordan algebras
论文作者
论文摘要
由Horn的log-majorization（单数值）的动机$ s（ab）\ usterset {log} {\ prec} {\ prec} s（a）*s（b）$和相关的弱 - 莫约尔不平等$ s（ab）\ underset {w} {w} {w} $λ（\ sqrt {a} b \ sqrt {a}）\ underSet {log} {\ prec} {\ prec}λ（a）*λ（b）$（b）$ for -seidefinite矩阵和$λ（$λ（| a \ circe b |） （Hermitian）矩阵，其中$ a \ circ b $表示$ a $和$ b $和$*$的Jordan产品表示$ r^n $中的componentwise产品。在本文中，我们将这些不平等扩展到欧几里得约旦代数的设置，形式为$λ\ big（p _ {\ sqrt {a}}}（a}}（b）\ big）\ big）\ big）\ underSet {log} {\ prec} {\ prec} {\ prec} {\ prec} { b | \ big）\ underSet {w} {\ prec}λ（| a |）*λ（| b |）$用于所有$ a $ and $ b $，其中$ p_u $和$ p_u $和$λ（u）$表示，分别是egemenvalue表示元素$ u $ u $ u $ eigenvalue vectory。我们还描述了$λ（| a \ bullet b |）的形式的不等式{在应用程序的形式中，我们证明了广义的Hölder类型不等式$ || a \ circ b || _p \ leq || a || _r \，|| B || b || b || _s $，其中$ || x || x || x || _p：= ||λ（x） [1，\ infty] $ with $ \ frac {1} {p} = \ frac {1} {r} {r}+\ frac {1} {s} $。我们还提供了Lyapunov转换$ L_A $和$ p_a $的准确值，相对于两个频谱$ p $ -Norms。
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.
