General position of a projection and its image under a free unitary Brownian motion

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Given an orthogonal projection P and a free unitary Brownian motion Y = (Yₜ)ₜ ≥ ₀ in a W⋆-non commutative probability space such that Y and P are ⋆-free in Voiculescu's sense, the main result of this paper states that P and YₜPYₜ⋆ are in general position at any time t. To this end, we study the dynamics of the unitary operator SYₜSYₜ⋆ where S = 2P-1. More precisely, we derive a partial differential equation for the Herglotz transform of its spectral distribution, say μₜ. Then, we provide a flow on the interval [-1,1] in such a way that the Herglotz transform of μₜ composed with this flow is governed by both the Herglotz transforms of the initial (t=0) and the stationary (t = ∞) distributions. This fact allows to compute the weight that μₜ assigns to z=1 leading to the main result. As a by-product, the weight that the spectral distribution of the free Jacobi process assigns to x=1 follows after a normalization. In the last part of the paper, we use combinatorics of non crossing partitions in order to analyze the term corresponding to the exponential decay e-nt in the expansion of the n-th moment of SYₜSYₜ⋆.