Brown Measure Support and the Free Multiplicative Brownian Motion

The free multiplicative Brownian motion b_t is the large-N limit of Brownian motion B_t^N on the general linear group GL\(N;mathbb{C}\). We prove that the Brown measure for b_t---which is an analog of the empirical eigenvalue distribution for matrices---is supported on the closure of a certain domain Σ_t in the plane. The domain Σ_t was introduced by Biane in the context of the large-N limit of the Segal--Bargmann transform associated to GL\(N;mathbb{C}\). We also consider a two-parameter version, b_s,t: the large-N limit of a related family of diffusion processes on GL\(N;mathbb{C}\) introduced by the second author. We show that the Brown measure of b_s,t is supported on the closure of a certain planar domain Σ_s,t, generalizing Σ_t, introduced by Ho. In the process, we introduce a new family of spectral domains related to any operator in a tracial von Neumann algebra: the {\em L^p_n-spectrum} for ninmathbb{N} and pge 1, a subset of the ordinary spectrum defined relative to potentially-unbounded inverses. We show that, in general, the support of the Brown measure of an operator is contained in its L₂²-spectrum.