×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bercovici, H., Voiculescu, D.: Convolutions of measures with unbounded support. Indiana Univ. Math. J. 42(3), 733-773 (1993) · Zbl 0806.46070
[2] Bercovici, H., Voiculescu, D.: Superconvergence to the central limit and failure of the Cramér theorem for free random variables. Probab. Theory Related Fields 103(2), 215-222 (1995) · Zbl 0831.60036
[3] Bercovici, H., Voiculescu, D.: Regularity questions for free convolution. Nonselfadjoint operator algebras, operator theory, and related topics. Oper. Theory Adv. Appl. Birkhäuser, Basel, 104, 37-47 (1998) · Zbl 0927.46048
[4] Biane, P.: Processes with free increments. Math. Z. 227(1), 143-174 (1998) · Zbl 0902.60060
[5] Maassen, H.: Addition of freely independent random variables. J. Funct. Anal. 106(2), 409-438 (1992) · Zbl 0784.46047
[6] Nica, A., Speicher, R.: On the multiplication of free N-tuples of noncommutative random variables. Amer. J. Math. 118(4), 799-837 (1996) · Zbl 0856.46035
[7] Poltoratski, A.: Images of non-tangential sectors under Cauchy transforms. J. Anal. Math. 89, 385-395 (2003) · Zbl 1042.31002
[8] Voiculescu, D.: The analogues of entropy and of Fisher?s information measure in free probability theory. I. Comm. Math. Phys. 155(1), 411-440 (1993) · Zbl 0820.60001
[9] Voiculescu, D.: The coalgebra of the free difference quotient and free probability. Internat. Math. Res. Notices 2, 79-106 (2000) · Zbl 0952.46038
[10] Voiculescu, D.: Addition of certain noncommuting random variables. J. Funct. Anal. 66(3), 323-346 (1986) · Zbl 0651.46063
[11] Voiculescu, D.V., Dykema, K.J., Nica, A.: Free Random Variables. CRM Monograph Series, Vol. 1 Am. Math. Soc. Providence, RI, 1992 · Zbl 0795.46049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.