Free Holomorphic Functions on the Regular Polyball

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.