Bohr Inequalities on Noncommutative Polydomains

  • Gelu PopescuEmail author


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.


Multivariable operator theory Bohr’s inequality Noncommutative polydomain Free holomorphic function Berezin transform Weighted Fock space 

Mathematics Subject Classification

Primary 47A56 46L52 Secondary 32A38 47A63 



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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsThe University of Texas at San AntonioSan AntonioUSA

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