从优超到Horn不等式与酉轨道不等式
From Majorization to Horn and Unitary-Orbit Inequalities
- University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一个统一框架，将优超不等式提升为完整的Horn不等式（等价于酉轨道不等式），并应用于三类算子不等式，得到推广的酉轨道不等式。
AI中文摘要：
优超不等式通过前导部分和来编码谱比较，而完整的Horn不等式则刻画与Hermitian矩阵之和相关的所有可容许的选择特征值约束[Fulton, 2000]。我们发展了一个将优超不等式提升到完整Horn水平的一般框架，该水平等价于酉轨道不等式。主要成分是适用于选择特征值的坐标单调函数的广义Hersch--Zwahlen变分公式，以及一个Schubert几何提升原理，该原理将所需的Horn不等式归结为矩阵压缩上合适的标量化函数的凹性或凸性。作为例子，我们将该框架应用于三类算子不等式。首先，我们获得了由Aujla和Silva建立的凸函数的弱优超不等式的完整Horn扩展[Aujla and Silva, 2003]。其次，我们提升了由Carlen、Frank和Lieb以及Zhang得到的双变量正幂函数的凹性和凸性结果[Carlen, Frank, and Lieb, 2016; Zhang, 2020]。第三，我们建立了多变元测地线均值（包括加权Karcher均值和双变量几何均值）的乘法Horn不等式，扩展了Bourin和Hiai的行列式不等式[Bourin and Hiai, 2014]。所得结果给出了酉轨道不等式，推广了熟悉的弱优超、迹和行列式不等式。
英文摘要：
Majorization inequalities encode spectral comparisons through leading partial sums, whereas complete Horn inequalities capture all admissible selected-eigenvalue constraints associated with sums of Hermitian matrices [Fulton, 2000]. We develop a general framework for lifting majorization inequalities to this complete Horn level, which is equivalent to unitary-orbit inequalities. The main ingredients are a generalized Hersch--Zwahlen variational formula for coordinatewise monotone functions of selected eigenvalues and a Schubert-geometric lifting principle that reduces the desired Horn inequalities to the concavity or convexity of suitable scalarizer functions on matrix compressions. As examples we apply the framework to three families of operator inequalities. First, we obtain a complete Horn extension of the weak-majorization inequality for convex functions established by Aujla and Silva [Aujla and Silva, 2003]. Second, we lift the concavity and convexity results for two-variable positive-power functions due to Carlen, Frank, and Lieb and Zhang [Carlen, Frank, and Lieb, 2016; Zhang, 2020]. Third, we establish multiplicative Horn inequalities for multivariable geodesic means, including the weighted Karcher mean and the two-variable geometric mean, extending determinant-level inequalities of Bourin and Hiai [Bourin and Hiai, 2014]. The resulting statements yield unitary-orbit inequalities that generalize familiar weak-majorization, trace, and determinant inequalities.
