On the Nevanlinna problem -- Characterization of all Schur-Agler class solutions affiliated with a given kernel

Abstract

Given a domain $\Omega$ in ${\mathbb{C}}^m$, and a finite set of points $z_1,z_2,\dots ,z_n\in \Omega$ and $w_1,w_2,\dots ,w_n\in \mathbb{D}$ (the open unit disc in the complex plane), the \textit{Pick interpolation problem} asks when there is a holomorphic function $f:\Omega \to \overline{\mathbb{D}}$ such that $f(z_i)=w_i,1\le i\le n$. Pick gave a condition on the data $\{z_i,w_i:1\le i\le n\}$ for such an $interpolant$ to exist if $\Omega =\mathbb{D}$. Nevanlinna characterized all possible functions $f$ that \textit{interpolate} the data. We generalize Nevanlinna's result to a domain $\Omega$ in ${\mathbb{C}}^m$ admitting holomorphic test functions when the function $f$ comes from the Schur-Agler class and is affiliated with a certain completely positive kernel. The Schur class is a naturally associated Banach algebra of functions with a domain. The success of the theory lies in characterizing the Schur class interpolating functions for three domains - the bidisc, the symmetrized bidisc and the annulus - which are affiliated to given kernels.

arXiv admin note: text overlap with arXiv:1812.00147, Accepted in Studia Mathematica