几何凹函数与几何凸函数的Araki-Lieb-Thirring不等式
An Araki-Lieb-Thirring inequality for geometrically concave and geometrically convex functions
摘要：
对于正定矩阵$A$和$B$，Araki-Lieb-Thirring不等式对于分数幂$$λ(A^t B^t) \prec_{w(\log)} λ^t(AB), \quad 0<t\le 1,$$等于特征值对数次优控制关系，而对于$t\ge1$，逆不等式成立.在本文中，我推广这个不等式，取代分数权力$x^t$的一个更大的功能类。也就是说，一个连续的、非负的、几何凹函数$f$，其域$\dom(f)=[0,x_0)$对某个正的$x_0$（可能是无穷大）满足$$λ(f(A) f(B)) \prec_{w(\log)} f^2(λ^{1/2}(AB)),$$对所有半正定的$A$和谱在$\dom(f)$中的$B$，当且仅当$0\le xf'(x)\le f(x)$对所有$x\in\dom(f)$。逆不等式对于连续的、非负的、几何凸函数成立当且仅当它们对所有$x\in\dom(f)$都满足$xf'(x)\ge f(x)$。作为一个应用，我得到了一个互补的不等式的Golden-Thompson不等式。
英文摘要：
For positive definite matrices $A$ and $B$, the Araki-Lieb-Thirring inequality amounts to an eigenvalue log-submajorisation relation for fractional powers $$λ(A^t B^t) \prec_{w(\log)} λ^t(AB), \quad 0<t\le 1,$$ while for $t\ge1$, the reversed inequality holds. In this paper I generalise this inequality, replacing the fractional powers $x^t$ by a larger class of functions. Namely, a continuous, non-negative, geometrically concave function $f$ with domain $\dom(f)=[0,x_0)$ for some positive $x_0$ (possibly infinity) satisfies $$λ(f(A) f(B)) \prec_{w(\log)} f^2(λ^{1/2}(AB)),$$ for all positive semidefinite $A$ and $B$ with spectrum in $\dom(f)$, if and only if $0\le xf'(x)\le f(x)$ for all $x\in\dom(f)$. The reversed inequality holds for continuous, non-negative, geometrically convex functions if and only if they satisfy $xf'(x)\ge f(x)$ for all $x\in\dom(f)$. As an application I derive a complementary inequality to the Golden-Thompson inequality.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
