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Register a new userThe Brown measure of the sum of a self-adjoint element and an elliptic element
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Ching-Wei Ho
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We compute the Brown measure of the sum of a self-adjoint element and an elliptic element. We prove that the push-forward of this Brown measure of a natural map is the law of the free convolution of the self-adjoint element and the semicircle law; it is also a push-forward measure of the Brown measure of the sum of the self-adjoint element and a circular element by another natural map. We also study various asymptotic behaviors of this family of Brown measures as the variance of the elliptic element approaches infinity.
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We compute the Brown measure of $x_{0}+isigma_{t}$, where $sigma_{t}$ is a free semicircular Brownian motion and $x_{0}$ is a freely independent self-adjoint element. The Brown measure is supported in the closure of a certain bounded region $Omega_{t}$ in the plane. In $Omega_{t},$ the Brown measure is absolutely continuous with respect to Lebesgue measure, with a density that is constant in the vertical direction. Our results refine and rigorize results of Janik, Nowak, Papp, Wambach, and Zahed and of Jarosz and Nowak in the physics literature. We also show that pushing forward the Brown measure of $x_{0}+isigma_{t}$ by a certain map $Q_{t}:Omega_{t}rightarrowmathbb{R}$ gives the distribution of $x_{0}+sigma_{t}.$ We also establish a similar result relating the Brown measure of $x_{0}+isigma_{t}$ to the Brown measure of $x_{0}+c_{t}$, where $c_{t}$ is the free circular Brownian motion.
Let $x_0$ be a possibly-unbounded self-adjoint random variable, $tildesigma_alpha$ and $sigma_beta$ are semicircular variables with variances $alphageq 0$ and $beta>0$ respectively (when $alpha = 0$, $tildesigma_alpha = 0$). Suppose $x_0$, $sigma_alpha$, and $tildesigma_beta$ are all freely independent. We compute the Brown measure of $x_0+tildesigma_alpha+isigma_beta$, extending the previous computations which only work for bounded self-adjoint random variable $x_0$. The Brown measure in this unbounded case has the same structure as in the bounded case; it has connections to the free convolution $x_0+sigma_{alpha+beta}$. We also compute the example where $x_0$ is Cauchy-distributed.
Let $x_0$ be a self-adjoint random variable and $c_t$ be a free circular Brownian motion, freely independent from $x_0$. We use the Hamilton--Jacobi method to compute the Brown measure $rho_t$ of $x_0+c_t$. The Brown measure is absolutely continuous with a density that is emph{constant along the vertical direction} in the support of $rho_t$. The support of the Brown measure of $x_0+c_t$ is related to the subordination function of the free additive convolution of $x_0+s_t$, where $s_t$ is the free semicircular Brownian motion, freely independent from $x_0$. Furthermore, the push-forward of $rho_t$ by a natural map is the law of $x_0+s_t$. Let $u$ be a unitary random variable and $b_t$ is the free multiplicative Brownian motion freely independent from $u$, we compute the Brown measure $mu_t$ of the free multiplicative Brownian motion $ub_t$, extending the recent work by Driver--Hall--Kemp. The measure is absolutely continuous with a density of the special form [frac{1}{r^2}w_t(theta)] in polar coordinates in its support. The support of $mu_t$ is related to the subordination function of the free multiplicative convolution of $uu_t$ where $u_t$ is the free unitary Brownian motion free independent from $u$. The push-forward of $mu_t$ by a natural map is the law of $uu_t$. In the special case that $u$ is Haar unitary, the Brown measure $mu_t$ follows the emph{annulus law}. The support of the Brown measure of $ub_t$ is an annulus with inner radius $e^{-t/2}$ and outer radius $e^{t/2}$. The density in polar coordinates is given by [frac{1}{2pi t}frac{1}{r^2}] in its support.
Let $mathfrak{g}$ be a simple Lie algebra of rank $r$ over $mathbb{C}$, $mathfrak{h} subset mathfrak{g}$ a Cartan subalgebra. We construct a family of $r$ commuting Hermitian operators acting on $mathfrak{h}$ whose eigenvalues are equal to the coordinates of the eigenvectors of the Cartan matrix of $mathfrak{g}$.
We compute the deficiency spaces of operators of the form $H_A{hat{otimes}} I + I{hat{otimes}} H_B$, for symmetric $H_A$ and self-adjoint $H_B$. This enables us to construct self-adjoint extensions (if they exist) by means of von Neumanns theory. The structure of the deficiency spaces for this case was asserted already by Ibort, Marmo and Perez-Pardo, but only proven under the restriction of $H_B$ having discrete, non-degenerate spectrum.
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