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Atoms and regularity for measures in a partially defined free convolution semigroup

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Abstract.

Consider a Borel probability measure μ on the real line, and denote by {μ t : t≥1} the free additive convolution semigroup defined by Nica and Speicher. We show that the singular part of μ t is purely atomic and the density of μ t is locally analytic, provided that t > 1. The main ingredient is a global inversion theorem for analytic functions on a half plane.

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Authors and Affiliations

  1. Institute of Mathematics, Romanian Academy, 1-764, 70700, Bucharest, Romania

    S. T. Belinschi

  2. Mathematics Department, Indiana University, Bloomington, IN, 47405, USA

    H. Bercovici

Authors
  1. S. T. Belinschi
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  2. H. Bercovici
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Corresponding author

Correspondence to H. Bercovici.

Additional information

Mathematics Subject Classification (2000): 46L54, 30A99

Supported in part by a grant from the National Science Foundation.

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Belinschi, S., Bercovici, H. Atoms and regularity for measures in a partially defined free convolution semigroup. Math. Z. 248, 665–674 (2004). https://doi.org/10.1007/s00209-004-0671-y

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  • DOI: https://doi.org/10.1007/s00209-004-0671-y

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