论文标题
在欧几里得约旦代数上线性地图的弱弱劳化不平等
A pointwise weak-majorization inequality for linear maps over Euclidean Jordan algebras
论文作者
论文摘要
鉴于欧几里得约旦代数的线性地图$ t $等级$ n $，我们考虑了所有非负矢量$ q $ in $ r^n $中的$ q $，而减少的组件减少了满足弱点的不平等不平等$λ（| t（x）特征值地图和$*$表示$ r^n $中的componentwise产品。关于弱化的顺序，我们显示了该集合中最小值的存在。当$ t $是一个正图时，最小值向量被证明是$ t（e）$和$ t^*（e）$的特征值矢量的联接（按弱化的顺序），其中$ e $是代数的单位元素。这些结果类似于巴帕特的结果，在所有$ n \ times n $复杂矩阵的设置中，具有单数值图代替特征值图。他们还扩展了道，钟和高达的两个最新结果，证明是二次表示和Schur产品诱导的转换的结果。作为应用程序，我们提供了相对于光谱规范的一般线性图的规范的估计。
Given a linear map $T$ on a Euclidean Jordan algebra of rank $n$, we consider the set of all nonnegative vectors $q$ in $R^n$ with decreasing components that satisfy the pointwise weak-majorization inequality $λ(|T(x)|)\underset{w}{\prec}q*λ(|x|)$, where $λ$ is the eigenvalue map and $*$ denotes the componentwise product in $R^n$. With respect to the weak-majorization ordering, we show the existence of the least vector in this set. When $T$ is a positive map, the least vector is shown to be the join (in the weak-majorization order) of eigenvalue vectors of $T(e)$ and $T^*(e)$, where $e$ is the unit element of the algebra. These results are analogous to the results of Bapat, proved in the setting of the space of all $n\times n$ complex matrices with singular value map in place of the eigenvalue map. They also extend two recent results of Tao, Jeong, and Gowda proved for quadratic representations and Schur product induced transformations. As an application, we provide an estimate on the norm of a general linear map relative to spectral norms.
