Complex Analysis and Operator Theory

, Volume 12, Issue 7, pp 1617–1635 | Cite as

Free Holomorphic Functions on the Regular Polyball

  • Maofa Wang
  • Jian HuEmail author


In this paper, we study free holomorphic functions on the regular polyball, which was recently introduced by Popescu (Adv Math 279:104–158, 2015, Trans Am Math Soc 368:4357–4416, 2016). The purpose of this paper is to continue the line of Popescu to develop a theory of free holomorphic functions. Some results from the classical holomorphic function theory have free analogues in this noncommutative setting. In particular, we prove the maximum principle, Weierstrass and Montel type theorems for free holomorphic functions. As an application, we construct a metric on \(\mathrm{Hol}(\mathbf{B_n}(\mathcal {H}))\) such that it becomes a complete space.


Fock space Regular polyball Noncommutative Berezin transform Free holomorphic function 

Mathematics Subject Classification

Primary 47B33 Secondary 32A35 32A36 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsWuhan UniversityWuhanChina

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