#### 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 $R{U}_{t}S{U}_{t}^{\ast }$, where $R,S$ are two symmetries and ${\left({U}_{t}\right)}_{t\ge 0}$ is a free unitary Brownian motion, freely independent from $\left\{R,S\right\}$. In particular, for nonnull traces of $R$ and $S$, we prove that the spectral measure of $R{U}_{t}S{U}_{t}^{\ast }$ possesses two atoms at $±1$ and an ${L}^{\infty }$-density on the unit circle $\mathbb{T}$ for every $t>0$. Next, via a Szegő-type transformation of this law, we obtain a full description of the spectral distribution of $P{U}_{t}Q{U}_{t}^{\ast }$ beyond the case where $\tau \left(P\right)=\tau \left(Q\right)=\frac{1}{2}$. 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