Brown measure support and the free multiplicative Brownian motion
Brown measure support and the free multiplicative Brownian motion
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DOI：
10.1016/j.aim.2019.106771
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发表时间：
2018-09
影响因子：
1.7
通讯作者：
B. Hall;Todd Kemp
中科院分区：
文献类型：
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作者：
B. Hall;Todd Kemp
自由乘法布朗运动B t是布朗运动B tN在一般线性群GL（N; C）上的大N极限.我们证明了布朗测度的B t-这是一个模拟的经验特征值分布的矩阵-是支持的封闭的某个域的平面上。定义域是由Biane在与GL（N; C）相关的Segal-Bargmann变换的大N极限的上下文中引入的。我们还考虑了一个两参数的版本，B s，t：GL（N; C）上的一个相关的家庭的扩散过程的大N极限由第二作者介绍。证明了B s，t的Brown测度是由Ho引入的一个平面区域s，t的闭包所支持的，推广了Ho引入的Brown测度.在此过程中，我们引入了一类新的谱域：Lpn-谱（n∈ N，p≥ 1），它是相对于位势无界逆定义的普通谱的子集.我们表明，在一般情况下，布朗测度的运营商的支持是包含在其L2 2-谱。
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; 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; 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; 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 L p n-spectrum for n∈ N and p≥ 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.