On the matrix range of random matrices
On the matrix range of random matrices
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DOI：
10.7900/jot.2019dec04.2277
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发表时间：
2019-11
影响因子：
0.8
通讯作者：
Malte Gerhold;O. Shalit
中科院分区：
文献类型：
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作者：
Malte Gerhold;O. Shalit
这篇文章讨论了一个简单的问题：\textit{典型的随机矩阵范围是什么样的？}我们一方面研究算子元组的各种收敛模式之间的关系，另一方面研究矩阵范围相对于 Hausdorff 度量的连续性。特别是，我们证明了生成 C* 代数连续域的元组的矩阵范围是连续的，因为每个级别在 Hausdorff 度量中都是连续的。使用这一观察结果以及矩阵系综分布强收敛的已知结果，我们确定了独立 Wigner 或 Haar 系综的矩阵范围收敛到的极限矩阵范围。
This note treats a simple minded question: \textit{what does a typical random matrix range look like?} We study the relationship between various modes of convergence for tuples of operators on the one hand, and continuity of matrix ranges with respect to the Hausdorff metric on the other. In particular, we show that the matrix range of a tuple generating a continuous field of C∗-algebras is continuous in the sense that every level is continuous in the Hausdorff metric. Using this observation together with known results on strong convergence in distribution of matrix ensembles, we identify the limit matrix ranges to which the matrix ranges of independent Wigner or Haar ensembles converge.