Holomorphic functions on the symmetrized bidisk - realization, interpolation and extension

TL;DR

This paper develops new theorems for holomorphic functions on the symmetrized bidisk, including realization, interpolation, and extension results, advancing the understanding of function theory on this domain.

Contribution

It introduces a realization formula, a Nevanlinna-Pick interpolation theorem, and an extension characterization specifically for the symmetrized bidisk.

Findings

Realization formula for bounded holomorphic functions on the symmetrized bidisk.

Nevanlinna-Pick interpolation theorem with explicit formula for the symmetrized bidisk.

Characterization of subsets allowing norm-preserving holomorphic extensions.

Abstract

There are three new things in this paper about the open symmetrized bidisk $\mathbb G = \{(z_1+z_2, z_1z_2) : |z_1|, |z_2| < 1\}$. They are motivated in the Introduction. In this Abstract, we mention them in the order in which they will be proved. \begin{enumerate} \item The Realization Theorem: A realization formula is demonstrated for every $f$ in the norm unit ball of $H^\infty(\mathbb G)$. \item The Interpolation Theorem: Nevanlinna-Pick interpolation theorem is proved for data from the symmetrized bidisk and a specific formula is obtained for the interpolating function. \item The Extension Theorem: A characterization is obtained of those subsets $V$ of the open symmetrized bidisk $\mathbb G$ that have the property that every function $f$ holomorphic in a neighbourhood of $V$ and bounded on $V$ has an $H^\infty$-norm preserving extension to the whole of $\mathbb G$. \end{enumerate}