Spectral distribution of the free Jacobi process, revisited

Hamdi, Tarek

doi:10.2140/apde.2018.11.2137

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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 $R U_{t} S U_{t}^{*}$ , where $R, S$ are two symmetries and ${(U_{t})}_{t \geq 0}$ 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 $R U_{t} S U_{t}^{*}$ possesses two atoms at $\pm 1$ and an $L^{\infty}$ -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 $P U_{t} Q U_{t}^{*}$ beyond the case where $τ (P) = τ (Q) = \frac{1}{2}$ . 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

Vol. 11, No. 8, 2018