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Free Holomorphic Functions on the Noncommutative Polydomains and Universal Models

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In this paper, we study free holomorphic functions on the noncommutative polydomains \(\mathbf{D_f^m}(\mathcal {H})\), which were recently introduced by Popescu (J Funct Anal 265(10):2500–2552, 2013; Adv Math 279:104–158, 2015; Trans Am Math 368:4357–4416, 2016). We obtain some characterizations of free holomorphic functions by noncommutative Berezin transforms and the universal models associated with the abstract noncommutative polydomains \(\mathbf{D_f^m}\). As consequences, we provide noncommutative multivariable analogues of several results from the classical holomorphic function theory such as: Wiener inequality, Weierstrass theorem, Montel theorem, Vitali theorem. Moreover, we also prove that the universal models are essentially unique.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, China

    Jian Hu & Maofa Wang

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  1. Jian Hu
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  2. Maofa Wang
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Correspondence to Jian Hu.

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This project was supported by NSFC (11271293).

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Hu, J., Wang, M. Free Holomorphic Functions on the Noncommutative Polydomains and Universal Models. Results Math 73, 99 (2018). https://doi.org/10.1007/s00025-018-0861-2

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