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Izv. RAN. Ser. Mat., 2016, Volume 80, Issue 6, Pages 127–140 (Mi izv8467)  

Radial parts of Haar measures and probability distributions on the space of rational matrix-valued functions

Yu. A. Neretinabcd

a University of Vienna
b State Scientific Center of the Russian Federation - Institute for Theoretical and Experimental Physics, Moscow
c Lomonosov Moscow State University
d Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: We consider the space $\mathcal C$ of conjugacy classes of the unitary group $\mathrm U(n+m)$ with respect to a smaller unitary group $\mathrm U(m)$. It is known that to every element of $\mathcal C$ we can canonically assign a rational matrix-valued function (the Livshits characteristic function) on the Riemann sphere. We find an explicit expression for the natural measure on $\mathcal C$ obtained as the push-forward of the Haar measure of $\mathrm U(n+m)$ in terms of characteristic functions.

Keywords: inner functions, characteristic functions, Haar measure, Cayley transform, random functions.

Funding Agency Grant Number
Russian Science Foundation 14-50-00150
This work was supported by the Russian Science Foundation under grant no. 14-50-00150.


DOI: https://doi.org/10.4213/im8467

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English version:
Izvestiya: Mathematics, 2016, 80:6, 1118–1130

Bibliographic databases:

UDC: 512.546.32+517.547.5+517.548.5
MSC: 47A48, 28C10, 15B52
Received: 28.10.2015

Citation: Yu. A. Neretin, “Radial parts of Haar measures and probability distributions on the space of rational matrix-valued functions”, Izv. RAN. Ser. Mat., 80:6 (2016), 127–140; Izv. Math., 80:6 (2016), 1118–1130

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