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In this paper, we derive explicit expressions for the moments and for themixed moments of the compression of a free unitary Brownian motion by a freeprojection. While the moments of this non-normal operator are readily derivedusing analytical or combinatorial methods, we only succeeded to derive itsmixed ones after solving a non-linear partial differential equation for theirtwo-variables generating function. Nonetheless, the combinatorics of noncrossing partitions lead to another expression of the lowest-order mixedmoment. We shall also give some interest in odd alternating moments. Inparticular, we derive a linear partial differential equation for theirgenerating function and discuss the combinatorial approach to these momentswhen the rank of the projection equals $1/2$.

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