Spectral distribution of the free Jacobi process, revisited

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 $\pm 1$ 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.