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

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.

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