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

Abstract

There are three new things in this paper about the open symmetrized bidisk $\mathbb{G}=\{(z_1+z_2,z_1{z}_2):|z_1|,|z_2|<1\}$. 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}

19 pages, a non-trivial error fixed

References in corpus (1)

• Test Functions, Kernels, Realizations and Interpolation

Cited by in corpus (6)

• Toeplitz operators on the symmetrized bidisc
• Function theory on quotient domains related to the polydisc
• Distinguished varieties and the Nevanlinna-Pick interpolation problem on the symmetrized bidisk
• de Branges-Rovnyak spaces which are complete Nevanlinna-Pick spaces
• Realization of functions on the symmetrized bidisc
• Holomorphic functions on the symmetrized bidisk II -- Interpolating sequences and the Toeplitz corona theorem