Vol. 11, No. 8, 2018

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Spectral distribution of the free Jacobi process, revisited

Tarek Hamdi

Vol. 11 (2018), No. 8, 2137–2148
DOI: 10.2140/apde.2018.11.2137
Abstract

We obtain a description for the spectral distribution of the free Jacobi process for any initial pair of projections. This result relies on a study of the unitary operator RUtSUt, where R,S are two symmetries and (Ut)t0 is a free unitary Brownian motion, freely independent from {R,S}. In particular, for nonnull traces of R and S, we prove that the spectral measure of RUtSUt possesses two atoms at ± 1 and an L-density on the unit circle T for every t > 0. Next, via a Szegő-type transformation of this law, we obtain a full description of the spectral distribution of PUtQUt beyond the case where τ(P) = τ(Q) = 1 2. Finally, we give some specializations for which these measures are explicitly computed.

Keywords
free Jacobi process, free unitary Brownian motion, multiplicative convolution, spectral distribution, Herglotz transform, Szegő transformation
Mathematical Subject Classification 2010
Primary: 42B37, 46L54
Milestones
Received: 23 November 2017
Revised: 20 March 2018
Accepted: 19 April 2018
Published: 6 June 2018
Authors
Tarek Hamdi
Department of Management Information Systems
College of Business Administration
Qassim University
Buraydah
Saudi Arabia
Laboratoire d’Analyse Mathématiques et Applications LR11ES11
Université de Tunis El-Manar
Tunis
Tunisia