Matrix-valued Aleksandrov–Clark measures and Carathéodory angular derivatives

This paper deals with families of matrix-valued Aleksandrov--Clark measures $\{\boldsymbol{\mu}^\alpha\}_{\alpha\in\mathcal{U}(n)}$, corresponding to purely contractive $n\times n$ matrix functions $b$ on the unit disc complex plane. We do not make other apriori assumptions $b$. In particular, may be non-inner and/or non-extreme. The study such is mainly motivated from applications unitary finite rank perturbation theory. A description absolutely continuous parts $\boldsymbol{\mu}^\alpha$ a rather straightforward generalization well-known results for scalar case ($n=1$). and proofs singular are more complicated than in case, constitute main focus this paper. discuss Aronszajn--Donoghue theory concerning Clark measures, as well Carath\'{e}odory angular derivatives their connections atoms $\boldsymbol{\mu}^\alpha$. These far being extensions case: new phenomena specific appear here. New ideas, including notion directionality, required statements proofs.