Matrix-valued Aleksandrov–Clark measures and Carathéodory angular derivatives
Autor: | Constanze Liaw, Sergei Treil, Robert T. W. Martin |
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Rok vydání: | 2021 |
Předmět: |
Pure mathematics
Mathematics - Complex Variables 010102 general mathematics Scalar (mathematics) 30H10 30H05 47B32 47B38 46E22 Absolute continuity Mathematical proof 01 natural sciences Unitary state Mathematics - Functional Analysis Matrix function 0103 physical sciences A priori and a posteriori 010307 mathematical physics 0101 mathematics Complex plane Analysis Mathematics |
Zdroj: | Journal of Functional Analysis. 280:108830 |
ISSN: | 0022-1236 |
Popis: | 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 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 $\boldsymbol{\mu}^\alpha$ 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 $\boldsymbol{\mu}^\alpha$ 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 Carath\'{e}odory angular derivatives of matrix-valued functions and their connections with atoms of $\boldsymbol{\mu}^\alpha$. 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. Comment: 28 pages; v2 has updated bibliography |
Databáze: | OpenAIRE |
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