On star moments of the compression of the free unitary Brownian motion by a free projection

Abstract

In this paper, we derive explicit expressions for some classes of ⋆-moments of a free unitary Brownian motion compressed by a free projection, using various methods. While the moments of this nonnormal operator are readily derived through analytical or combinatorial methods, we only succeeded to derive its mixed ones after solving a nonlinear partial differential equation (pde) for their generating function. We shall also give some interest in odd alternating moments. In particular, we derive a linear pde for their generating function which we solve when the rank of the projection equals~1/2.