Abstract This paper deals with families of matrix-valued Aleksandrov–Clark measures { μ α } α ∈ U ( n ) , corresponding to purely contractive n × 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 μ 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 μ 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 Caratheodory angular derivatives of matrix-valued functions and their connections with atoms of μ 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.

中文翻译：

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矩阵值 Aleksandrov-Clark 测度和 Carathéodory 角导数

摘要 本文涉及矩阵值的 Aleksandrov-Clark 测度族 { μ α } α ∈ U ( n ) ，对应于复平面单位圆盘上的纯收缩 n × n 矩阵函数 b。我们不对 b 做其他先验假设。特别地，b 可以是非内部的和/或非极端的。这类族的研究主要是从应用到酉有限秩扰动理论中得到的。μ α 的绝对连续部分的描述是对标量情况 (n = 1) 的众所周知的结果的相当直接的概括。矩阵值 μ α 奇异部分的结果和证明比标量情况更复杂，是本文的主要关注点。我们讨论关于克拉克测度奇异部分的矩阵值 Aronszajn-Donoghue 理论，以及矩阵值函数的 Caratheodory 角导数及其与 μ α 原子的联系。这些结果远非标量情况的直接扩展：这里出现了特定于矩阵值情况的新现象。陈述和证明需要新的想法，包括方向性的概念。