Spectral distribution of the free Jacobi process, revisited

TL;DR

This paper characterizes the spectral distribution of the free Jacobi process for any initial projections, revealing atoms and densities, and extends previous results through a Szeg\

Contribution

It provides a comprehensive description of the spectral distribution of the free Jacobi process for arbitrary initial projections, including explicit computations in special cases.

Findings

Spectral measure has two atoms at  and  with an $L^\u2208$-density on .

Full spectral distribution described via a Szeg\

Explicit measures computed for specific cases.

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 $RU_tSU_t^*$ where $R,S$ are two symmetries and $U_t$ a free unitary Brownian motion, freely independent from $\{R,S\}$. In particular, for non-null traces of $R$ and $S$, we prove that the spectral measure of $RU_tSU_t^*$ possesses two atoms at $\pm1$ and an $L^\infty$-density on the unit circle $\mathbb{T}$, for every $t>0$. Next, via a Szeg\H{o} type transform of this law, we obtain a full description of the spectral distribution of $PU_tQU_t^*$ beyond the $\tau(P)=\tau(Q)=1/2$ case. Finally, we give some specializations for which these measures are explicitly computed.