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Atoms and regularity for measures in a partially defined free convolution semigroup. (English) Zbl 1065.46045
Summary: Consider a Borel probability measure $$\mu$$ on the real line, and denote by $$\{\mu_t : t \geq 1\}$$ the free additive convolution semigroup defined by A. Nica and R. Speicher [Am. J. Math. 118, No. 4, 799–837 (1996; Zbl 0856.46035)]. We show that the singular part of $$\mu_t$$ is purely atomic and the density of $$\mu_t$$ is locally analytic, provided that $$t>1$$. The main ingredient is a global inversion theorem for analytic functions on a half plane.

MSC:
 46L54 Free probability and free operator algebras 30A99 General properties of functions of one complex variable 30B40 Analytic continuation of functions of one complex variable 30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane 31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
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References:
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