Spectral results for free random variables

TL;DR

The paper introduces a sharp criterion: the real-analytic extendability of the derivative of the regularized log potential, $\\partial S/\\partial \\varepsilon$, at $\\varepsilon=0$ characterizes when a complex number lies outside the spectrum of an operator in a tracial von Neumann algebra. It then leverages this criterion, together with the PDE approach for Brown measures, to prove that in many free-additive and free-multiplicative models the spectrum coincides with the Brown measure’s support. These results are applied to sums and products involving circular, elliptic, and free multiplicative Brownian motions, establishing spectral-equality results and explicit spectral domains $\\Sigma_t$ and their push-forwards $\\Phi_{t,\\gamma}$ and $\\Psi_{t,\\gamma}$. The work clarifies when the Brown measure’s support fully determines the spectrum and supplies tools to compute spectra in several non-normal free-probability settings. It also provides concrete examples showing the limits of spectrum–Brown-support equality in certain non-self-adjoint or non-normal constructions. Overall, the paper deepens the connection between analytic regularizations, PDE methods, and spectral theory in free probability.

Abstract

Let $(\mathcal{A},\mathrm{tr})$ be a von Neumann algebra with a faithful, normal trace $\mathrm{tr}:\mathcal{A}\rightarrow\mathbb{C}.$ For each $a\in\mathcal{A},$ define \[ S(λ,\varepsilon)=\mathrm{tr}[\log((a-λ)^{\ast}(a-λ)+\varepsilon)],\quadλ\in\mathbb{C},~\varepsilon>0, \] so that the limit as $\varepsilon\rightarrow0^{+}$ of $S$ is the log potential of the Brown measure of $a.$ Suppose that for a fixed $λ\in\mathbb{C},$ the function \[ \varepsilon\mapsto\frac{\partial S}{\partial\varepsilon}(λ,\varepsilon)=\mathrm{tr}[((a-λ)^{\ast}(a-λ)+\varepsilon )^{-1}] \] admits a real analytic extension to a neighborhood of $0$ in $\mathbb{R}.$ Then we will show that $λ$ is outside the spectrum of $a.$ We will apply this result to several examples involving circular and elliptic elements, as well as free multiplicative Brownian motions. In most cases, we will show that the spectrum of the relevant element $a$ coincides with the support of its Brown measure.

Figures (7)

• Figure 1: The region $\Sigma_{t}$ from DHK for $t=4$ (left), and a detail thereof (right).
• Figure 2: Density plot of the Brown measure of $x+c_{t}$, where $x$ is self-adjoint with $\mu_{x}=\frac{1}{2}(\delta_{-1}+\delta_{1}),$ with $t=1.$
• Figure 3: Density plot of the Brown measure of $x+c_{t},$ where $x$ is unitary and $\mu_{x}$ is supported at the third roots of unity, with equal masses, and $t=1.$
• Figure 4: Density plot of the Brown measure of $x+g_{t,\gamma}$ with $x$ self-adjoint and $\mu_{x}=\frac{1}{2}(\delta_{-1}+\delta_{1}),$ with $t=1$ and $\gamma=-i/2.$
• Figure 5: Density plot of Brown measure of $ub_{t},$ where $u$ is unitary with $\mu_{u}$ supported at the fourth roots of unity, with equal mass, for $t=1.$

Theorems & Definitions (68)

• Proposition 2.1
• Example 2.2: Haagerup--Larsen
• Proposition 2.3
• proof
• Remark 2.4
• Theorem 2.5: Zhong
• Theorem 2.6
• Remark 2.7
• Lemma 2.8
• proof