Free holomorphic functions on the unit ball of Bn(H)B(H)n, II

In this paper we continue the study of free holomorphic functions on the noncommutative ball[B(H)n]1:={(X1,…,Xn)∈B(H)n:‖X1X1∗+⋯+XnXn∗‖1/2<1}, where B(H)B(H) is the algebra of all bounded linear operators on a Hilbert space HH, and n=1,2,…n=1,2,… or n=∞n=∞. Several classical results from complex analysis have free analogues in our noncommutative setting. We prove a maximum principle, a Naimark type representation theorem, and a Vitali convergence theorem, for free holomorphic functions with operator-valued coefficients. We introduce the class of free holomorphic functions with the radial infimum property and study it in connection with factorizations and noncommutative generalizations of some classical inequalities obtained by Schwarz and Harnack. The Borel–Carathéodory theorem is extended to our noncommutative setting. Using a noncommutative generalization of Schwarz's lemma and basic facts concerning the free holomorphic automorphisms of the noncommutative ball [Bn(H)]1[B(H)n]1, we obtain an analogue of Julia's lemma for free holomorphic functions F:[Bn(H)]1→[Bm(H)]1F:[B(H)n]1→[B(H)m]1. We also obtain Pick–Julia theorems for free holomorphic functions with operator-valued coefficients. We provide a noncommutative generalization of a classical inequality due to Lindelöf, which turns out to be sharper then the noncommutative von Neumann inequality. Finally, we introduce a pseudohyperbolic metric on [Bn(H)]1[B(H)n]1 which is invariant under the action of the free holomorphic automorphism group of [Bn(H)]1[B(H)n]1 and turns out to be a noncommutative extension of the pseudohyperbolic distance on BnBn, the open unit ball of CnCn. In this setting, we obtain a Schwarz–Pick type lemma. We also provide commutative versions of these results for operator-valued multipliers of the Drury–Arveson space.