Given a linear map T on a Euclidean Jordan algebra of rank n, we consider the set of all nonnegative vectors q in ${\mathcal{R}}^n$ with decreasing components that satisfy the pointwise weak-majorization inequality $\lambda \left(|T\right(x\left)|\right)\underset{w}{\prec }q\ast \lambda \left(|x|\right)$, where λ is the eigenvalue map and $\ast$ denotes the componentwise product in ${\mathcal{R}}^n$. 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 $T\left(e\right)$ and $T^{\ast }\left(e\right)$, where e is the unit element of the algebra. These results are analogous to the results of Bapat [Majorization and singular values. III. Linear Algebra Appl. 1991;145:59–70] on singular values. We also extend two recent results of Tao et al. [Some log and weak majorization inequalities in Euclidean Jordan algebras. 2020. arXiv:2003.12377v2] 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.

给出的线性地图牛逼秩的欧几里德若代数ñ，我们认为集中的所有非负向量q在${\mathcal{[R}}^{ñ}$ 满足点弱弱不等式的递减分量 $\lambda （|Ť（X）|）\underset{w}{\prec }q\ast \lambda （|X|）$，其中λ是特征值图，而$\ast$ 表示 ${\mathcal{[R}}^{ñ}$。关于弱专业化排序，我们显示了该集合中最小向量的存在。当T是一个正图时，最小向量显示为特征值向量的连接（按弱主次排序）。$Ť（Ë）$ 和 ${Ť}^{\ast }（Ë）$，其中e是代数的单位元素。这些结果类似于Bapat [专业化和奇异值。三，线性代数应用 1991; 145：59-70]。我们还扩展了Tao等人的两个最新结果。[欧几里得乔丹代数中的一些对数和弱的不等式。2020. arXiv：2003.12377v2]证明了二次表示和Schur积诱发的变换。作为一种应用，我们提供了相对于频谱范数的一般线性图范数的估计。