AI 中文总结
研究\(n\times n\)严格收缩矩阵\(T\)的自守Nelson扩张，通过该扩张可快速获其最小等距和酉扩张，与经典理论相连。还证明每个\(T\)的Nelson扩张有特别性质，如强纠缠特征值函数或与\(z\)无关的约化子空间，同时研究了相关矩阵乘积特征值情况。
AI 中文摘要
给定一个\(n\times n\)严格收缩矩阵\(T\)，\(T\)的（自守）Nelson扩张\(\widehat{T}\)是单位圆盘上的一种特定解析矩阵值函数，且\(\widehat{T}(0)=T\)。其构造为将矩阵提升为具有良好边界行为的矩阵值函数提供了一种方法。本文表明Nelson扩张能快速得到\(T\)的最小等距和酉扩张，与经典的Sz.-Nagy扩张理论自然相连。接着将自守Nelson扩张作为基本对象展开研究，证明每个\(T\)都有具有特别有用/有趣性质的Nelson扩张，如具有强纠缠特征值函数或有与\(z\)无关的约化子空间。过程中还研究了可逆矩阵与对角矩阵乘积何时有不同特征值。
英文摘要
Given an $n \times n$ strictly contractive matrix $T$, an (automorphic) Nelson dilation $\widehat{T}$ of $T$ is a certain type of analytic matrix-valued function on the unit disk with $\widehat{T}(0) = T$. Its construction gives a method for lifting a matrix to a matrix-valued function with nice boundary behavior, a trick that has proved useful in recent operator theoretic developments. In this paper, we show that Nelson dilations give a quick way to obtain the minimal isometric and unitary dilations of $T$ and thus, connect naturally to the classical Sz.-Nagy dilation theory. We then initiate the study of the automorphic Nelson dilations as a fundamental object in their own right and prove that every $T$ has Nelson dilations $\widehat{T}$ with particularly useful/interesting properties; for example, they either have strongly entangled eigenvalue functions or have reducing subspaces that are independent of $z$. Along the way, we examine when the product of an invertible matrix and a diagonal matrix has distinct eigenvalues.
Comments21 pages, 2 figures
