The free multiplicative Brownian motion $b_{ t}$ is the large-$N$ limit of Brownian motion $B_t^N$ on the general linear group $\mathrm{ GL}(N;\mathbb{ C})$. 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 $\Sigma_{ t}$ in the plane. The domain $\Sigma_t$ was introduced by Biane in the context of the large-$N$ limit of the Segal--Bargmann transform associated to $\mathrm{ GL}(N;\mathbb{ C})$. We also consider a two-parameter version, $b_{ s,t}$: the large-$N$ limit of a related family of diffusion processes on $\mathrm{ GL}(N;\mathbb{ C})$ 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 $\Sigma_{ s,t}$, generalizing $\Sigma_t$, 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 $n\in\mathbb{ N}$ and $p\ge 1$, 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 $L_2^2$-spectrum.
Categories: math.FA hep-th math-ph math.MP math.PR quant-ph