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Journal of Operator Theory

Volume 90, Issue 2, Autumn 2023  pp. 453-489.

The Brown measure of unbounded variables with free semicircular imaginary part

Authors:  Ching-Wei Ho
Author institution: Institute of Mathematics, Academia Sinica, Taipei 10617, Taiwan

Summary:  Let $x_0$ be an unbounded self-adjoint operator such that the Brown measure of $x_0$ exists in the sense of Haagerup and Schultz. Let $\widetilde\sigma_\alpha$ and $\sigma_\beta$ be semicircular variables with variances $\alpha\geqslant 0$ and $\beta>0$. Suppose $x_0$, $\sigma_\alpha$, and $\widetilde\sigma_\beta$ are free. We use the PDE method introduced by Driver, Hall and Kemp to compute the Brown measure of $x_0+\widetilde\sigma_\alpha+\mathrm{i}\sigma_\beta$, extending the recent work which assume $x_0$ is a bounded self-adjoint operator. The computation of the PDE relies on a characterization of the class of operators where the Brown measure exists. We also compute the example where $x_0$ is Cauchy-distributed.

Keywords: free probability, Brown measure, unbounded operator, Cauchy distribution, random matrices

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