论文标题
矩阵值Aleksandrov- c-Clark措施和CarathéodoryAngular衍生物
Matrix-valued Aleksandrov--Clark measures and Carathéodory angular derivatives
论文作者
论文摘要
本文介绍了矩阵值的aleksandrov--clark的家族$ \ {\boldsymbolμ^α\} _ {α\ in \ Mathcal {u}（u}（n）} $，对应于纯粹的合同$ n \ n \ n $ matrix $ b $ b $ b $ b $ b $ b $ b $ b。我们不对$ b $做出其他Apriori假设。特别是，$ b $可能是非inner和/或非限制的。从应用到统一有限等级扰动理论，对此类家庭的研究主要是引起的。 对$ \boldsymbolμ^α$的绝对连续部分的描述是对标量案例的众所周知结果的简单概括（$ n = 1 $）。 矩阵值$ \boldsymbolμ^α$的奇异部分的结果和证明比标量案例更为复杂，构成了本文的主要重点。我们讨论了有关Clark测度的奇异部分以及基质值函数的Carathéodory角衍生物及其与$ \boldsymbolμ^α$的原子的连接。这些结果远非标量案例的直接扩展：矩阵值案例特有的新现象出现在此处。在陈述和证明中需要新想法，包括方向性的概念。
This paper deals with families of matrix-valued Aleksandrov--Clark measures $\{\boldsymbolμ^α\}_{α\in\mathcal{U}(n)}$, corresponding to purely contractive $n\times n$ matrix functions $b$ on the unit disc of the complex plane. We do not make other apriori assumptions on $b$. In particular, $b$ may be non-inner and/or non-extreme. The study of such families is mainly motivated from applications to unitary finite rank perturbation theory. A description of the absolutely continuous parts of $\boldsymbolμ^α$ is a rather straightforward generalization of the well-known results for the scalar case ($n=1$). The results and proofs for the singular parts of matrix-valued $\boldsymbolμ^α$ are more complicated than in the scalar case, and constitute the main focus of this paper. We discuss matrix-valued Aronszajn--Donoghue theory concerning the singular parts of the Clark measures, as well as Carathéodory angular derivatives of matrix-valued functions and their connections with atoms of $\boldsymbolμ^α$. These results are far from being straightforward extensions from the scalar case: new phenomena specific to the matrix-valued case appear here. New ideas, including the notion of directionality, are required in statements and proofs.
