Abstract:We study freely infinitely divisible $R$-diagonal elements in the unbounded setting and Brown measures for free additive perturbations by such elements. This class includes circular elements, circular Cauchy elements, and other previously studied $R$-diagonal models. We construct examples and prove stability under several algebraic operations, including homogeneous noncommutative polynomials in bounded, freely independent elements from this class. Using results for general $R$-diagonal perturbations, together with several analytic estimates specific to freely infinitely divisible $R$-diagonal elements, we prove that, in the bounded case, the support of the Brown measure coincides with the spectrum, and we obtain a criterion for property (H) in this non-normal setting. Finally, we study the free convolution semigroup associated with the symmetrized law of the modulus and derive a Hamilton--Jacobi equation for the regularized logarithmic potential.
| Comments: | Preliminary version, 25 pages |
| Subjects: | Operator Algebras (math.OA); Probability (math.PR) |
| MSC classes: | 46L54 |
| Cite as: | arXiv:2605.25434 [math.OA] |
| (or arXiv:2605.25434v1 [math.OA] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25434 arXiv-issued DOI via DataCite (pending registration) |
From: Yu Kitagawa [view email]
[v1]
Mon, 25 May 2026 05:17:10 UTC (33 KB)
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