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Bohr Inequalities on Noncommutative Polydomains

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Abstract

The goal of this paper is to study the Bohr phenomenon in the setting of free holomorphic functions on noncommutative regular polydomains \(\mathbf{D_f^m}\), \(\mathbf{f}=(f_1,\ldots , f_k)\), generated by positive regular free holomorphic functions. These polydomains are noncommutative analogues of the scalar polydomains

$$\begin{aligned} {{\mathcal {D}}}_{f_1} ({{\mathbb {C}}})\times \cdots \times {{\mathcal {D}}}_{f_k}({{\mathbb {C}}}), \end{aligned}$$

where each \({{\mathcal {D}}}_{f_i}({{\mathbb {C}}})\subset {{\mathbb {C}}}^{n_i}\) is a certain Reinhardt domain generated by \(f_i\). We characterize the free holomorphic functions on \(\mathbf{D_f^m}\) in terms of the universal model of the polydomain and extend several classical results from complex analysis to our noncommutative setting. It is shown that the free holomorphic functions admit multi-homogeneous and homogeneous expansions as power series in several variables. With respect to these expansions, we introduce the Bohr radii \(K_{mh}(\mathbf{D_f^m})\) and \(K_{h}(\mathbf{D_f^m})\) for the noncommutative Hardy space \(H^\infty (\mathbf{D_{f,\text { rad}}^m})\) of all bounded free holomorphic functions on the radial part of \(\mathbf{D_f^m}\). Several well-known results concerning the Bohr radius associated with classes of bounded holomorphic functions are extended to our noncommutative multivariable setting.

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  1. Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX, 78249, USA

    Gelu Popescu

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  1. Gelu Popescu
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Correspondence to Gelu Popescu.

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Research supported in part by NSF Grant DMS 1500922.

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Popescu, G. Bohr Inequalities on Noncommutative Polydomains. Integr. Equ. Oper. Theory 91, 7 (2019). https://doi.org/10.1007/s00020-019-2505-7

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