SPECTRAL DISTRIBUTION OF THE FREE JACOBI PROCESS, REVISITED

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 RUt SU*(t), where R, S are two symmetries and (U-t)(t >= 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 RUt SU*(t) possesses two atoms at +/- 1 and an L-infinity-density on the unit circle T for every t > 0. Next, via a Szego-type transformation of this law, we obtain a full description of the spectral distribution of PU(t)QU*(t) beyond the case where tau(P) = tau(Q) = 1/2. Finally, we give some specializations for which these measures are explicitly computed.