Advertisement

Complex Analysis and Operator Theory

, Volume 13, Issue 1, pp 61–84 | Cite as

Some Geometric Properties of the Subordination Function Associated to an Operator-Valued Free Convolution Semigroup

  • Serban Teodor BelinschiEmail author
Article
  • 54 Downloads

Abstract

In his article On the free convolution with a semicircular distribution, Biane found very useful characterizations of the boundary values of the imaginary part of the Cauchy–Stieltjes transform of the free additive convolution of a probability measure on \(\mathbb {R}\) with a Wigner (semicircular) distribution. Biane’s methods were recently extended by Huang (Int Math Res Not 12:4269–4292, 2015) to measures which belong to the partial free convolution semigroups introduced by Nica and Speicher. This note further extends some of Biane’s methods and results to free convolution powers of operator-valued distributions and to free convolutions with operator-valued semicirculars. In addition, it investigates properties of the Julia–Carathéodory derivative of the subordination functions associated to such semigroups, extending certain results from a previous article of H. Bercovici and the author to operator-valued maps.

Keywords

Free noncommutative functions Free probability Operator-valued distributions Free convolution semigroups 

Mathematics Subject Classification

46L54 46E50 32H50 

Notes

Acknowledgements

I am very grateful to Hari Bercovici for having provided valuable feedback on an earlier version of this paper.

References

  1. 1.
    Abduvalieva, G., Kaliuzhnyi-Verbovetskyi, D.S.: Fixed point theorems for noncommutative functions. J. Math. Anal. Appl. 401, 436–446 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Akhieser, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Translated by N. Kemmer. Hafner Publishing Co., New York (1965)Google Scholar
  3. 3.
    Anshelevich, M., Belinschi, S.T., Février, M., Nica, A.: Convolution powers in the operator-valued context. Trans. Am. Math. Soc. 365(4), 2063–2097 (2013)CrossRefzbMATHGoogle Scholar
  4. 4.
    Belinschi, S.T.: A noncommutative Version of the Julia–Wolff–Carathéodory Theorem, vol 95. pp 541–566 (2017). doi: 10.1112/jlms.12021
  5. 5.
    Belinschi, S.T., Mai, T., Speicher, R.: Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem. J. Reine Angew. Math., (2015). doi: 10.1515/crelle-2014-0138
  6. 6.
    Belinschi, S.T., Popa, M., Vinnikov, V.: Infinite divisibility and a noncommutative Boolean-to-free Bercovici–Pata bijection. J. Funct. Anal. 262, 94–123 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Belinschi, S.T., Bercovici, H.: Partially defined semigroups relative to multiplicative free convolution. Int. Math. Res. Not. 2, 65–101 (2005). doi: 10.1155/IMRN.2005.65 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Belinschi, S.T., Nica, A.: On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution. Indiana Univ. Math. J. 57(4), 1679–1713 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Belinschi, S.T., Nica, A.: Free Brownian motion and evolution towards \(\boxplus \)-infinite divisibility for \(k\)-tuples. Int. J. Math. 20(3), 309–338 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bercovici, H., Voiculescu, D.: Free convolutions of measures with unbounded support. Indiana Univ. Math. J. 42(3), 733–773 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bercovici, H., Voiculescu, D.: Superconvergence to the central limit and failure of the Cramér theorem for free random variables. Probab. Theory Relat. Fields 103(2), 215–222 (1995)CrossRefzbMATHGoogle Scholar
  12. 12.
    Biane, Ph: On the free convolution with a semi-circular distribution. Indiana Univ Math. J. 46(3), 705–718 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Blackadar, B.: Operator Algebras. In: Theory of C\({}^*\)-Algebras and von Neumann Algebras. Encyclopaedia of Mathematical Sciences, vol. 122. Springer, Berlin (2006)Google Scholar
  14. 14.
    Curran, S.: Analytic Subordination for Free Compression. Preprint (2008). arXiv:0803.4227v2 [math.OA]
  15. 15.
    Dineen, S.: The Schwarz Lemma. Oxford University Press, Oxford (1989)zbMATHGoogle Scholar
  16. 16.
    Earle, C.J., Hamilton, R.S.: A fixed point theorem for Holomorphic mappings. In: Global Analysis (Proceedings of Symposia in Pure Mathematics, Vol. XVI, Berkeley, 1968), pp. 61–65. AMS, Rhode Island (1970)Google Scholar
  17. 17.
    Helton, J. W., Rashidi Far, R., Speicher, R.: Operator-valued semicircular elements: solving a quadratic matrix equation with positivity constraints. Int. Math. Res. Not. (2007). doi: 10.1093/imrn/rnm086
  18. 18.
    Huang, H.-W.: Supports of measures in a free additive convolution semigroup. Int. Math. Res. Not. 12, 4269–4292 (2015). doi: 10.1093/imrn/rnu064 MathSciNetzbMATHGoogle Scholar
  19. 19.
    Kaliuzhnyi-Verbovetskyi, D. S., Vinnikov, V.: Foundations of Free Noncommutative Function Theory. Mathematical Surveys and Monographs, vol. 199. American Mathematical Society, Providence (2014)Google Scholar
  20. 20.
    Nica, A., Speicher, R.: On the multiplication of free \(N\)-tuples of noncommutative random variables. Am. J. Math. 118(4), 799–837 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Popa, M., Vinnikov, V.: Non-commutative functions and non-commutative free Lévy–Hinçin formula. Adv. Math. 236, 131–157 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Shlyakhtenko, D.: Random Gaussian band matrices and freeness with amalgamation. Int. Math. Res. Not. 20, 1013–1025 (1996). doi: 10.1155/S1073792896000633 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shlyakhtenko, D.: Gaussian random band matrices and operator-valued free probability theory. In: Quantum Probability (Gdańsk, 1997), ser. Banach Center Publ., vol. 43, pp. 359–368. Warsaw: Polish Academy Science (1998)Google Scholar
  24. 24.
    Shlyakhtenko, D.: On free convolution powers. Indiana Univ. Math. J. 62, 91–97 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Speicher, R.: Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Memoirs of the American Mathematical Society Publication, vol. 132. (1998). doi: 10.1090/memo/0627
  26. 26.
    Voiculescu, D.: Addition of certain non-commutative random variables. J. Funct. Anal. 66, 323–346 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Voiculescu, D.: Limit laws for random matrices and free products. Invent. Math. 104, 201–220 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Voiculescu, D.V.: Operations on certain non-commutative operator-valued random variables. Astérisque 232, 243–275 (1995)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Voiculescu, D.: The coalgebra of the free difference quotient and free probability. Int. Math. Res. Not. 2, 79–106 (2000). doi: 10.1155/S1073792800000064 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Voiculescu, D.: Free analysis questions I: duality transform for the coalgebra of \(\partial _{X:B}\). Int. Math. Res. Not. 16, 793–822 (2004). doi: 10.1155/S1073792804132443 MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Voiculescu, D.V.: Free analysis questions. II: the Grassmannian completion and the series expansions at the origin. J. Reine Angew. Math. 645, 155–236 (2010)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.CNRS - Institut de Mathématiques de ToulouseToulouseFrance

Personalised recommendations