Free holomorphic functions and interpolation

Abstract

In this paper we obtain a noncommutative multivariable analogue of the classical Nevanlinna–Pick interpolation problem for analytic functions with positive real parts on the open unit disc. Given a function $$f : \Lambda \to \mathbb {C}$$ , where $$\Lambda$$ is an arbitrary subset of the open unit ball $$\mathbb{B}_n:=\{z\in \mathbb {C}^n: \|z\| < 1\}$$ , we find necessary and sufficient conditions for the existence of a free holomorphic function g with complex coefficients on the noncommutative open unit ball $$[B({\mathcal H})^n]_1$$ such that $${\rm Re} \ g \geq 0 \quad {\rm and} \quad g(z)=f(z),\quad z\in \Lambda,$$ where $$B({\mathcal H})$$ is the algebra of all bounded linear operators on a Hilbert space $${\mathcal H}$$ . The proof employs several results from noncommutative multivariable operator theory and a noncommutative Cayley transform (introduced and studied in the present paper) acting from the set of all free holomorphic functions with positive real parts to the set of all bounded free holomorphic functions. All the results of this paper are obtained in the more general setting of free holomorphic functions with operator-valued coefficients. As consequences, we deduce some results concerning operator-valued analytic interpolation on the unit ball $${\mathbb B}_n$$ .