Free Holomorphic Functions on Polydomains

Abstract: In this paper, we continue to develop the theory of free holomorphic functions on noncommutative regular polydomains. We find analogues of several classical results from complex analysis such as Abel theorem, Hadamard formula, Cauchy inequality, and Liouville theorem for entire functions, in our multivariable setting. We also provide a maximum principle and a Schwarz type lemma. These results are used to prove analogues of Weierstrass, Montel, and Vitali theorems for the algebra of free holomorphic functions on the regular polydomain, which turns out to be a complete metric space.