Geodesics, retracts, and the norm-preserving extension property in the symmetrized bidisc

TL;DR

This paper characterizes subsets of the symmetrized bidisc with the norm-preserving extension property, linking them to complex geodesics and retracts, and explores implications for operator theory and related domains.

Contribution

It provides a complete classification of algebraic subsets with the extension property in the symmetrized bidisc, revealing new sets with this property beyond holomorphic retracts.

Findings

Characterization of subsets with the extension property as geodesics or unions involving geodesics.

Identification of sets with the extension property in domains like the spectral ball, tetrablock, and pentablock.

Application to von Neumann-type inequalities for mbda-contractions.

Abstract

A set $V$ in a domain $U$ in $\mathbb{C}^n$ has the {\em norm-preserving extension property} if every bounded holomorphic function on $V$ has a holomorphic extension to $U$ with the same supremum norm. We prove that an algebraic subset of the {\em symmetrized bidisc} \[ G := \{(z+w,zw):|z|<1, |w| < 1 \} \] has the norm-preserving extension property if and only if it is either a singleton, $G$ itself, a complex geodesic of $G$, or the union of the set $\{(2z,z^2): |z|<1\}$ and a complex geodesic of degree $1$ in $G$. We also prove that the complex geodesics in $G$ coincide with the nontrivial holomorphic retracts in $G$. Thus, in contrast to the case of the ball or the bidisc, there are sets in $G$ which have the norm-preserving extension property but are not holomorphic retracts of $G$. In the course of the proof we obtain a detailed classification of the complex geodesics in $G$ modulo automorphisms of $G$. We give applications to von Neumann-type inequalities for $\Gamma$-contractions (that is, commuting pairs of operators for which the closure of $G$ is a spectral set) and for symmetric functions of commuting pairs of contractive operators. We find three other domains that contain sets with the norm-preserving extension property which are not retracts: they are the spectral ball of $2\times 2$ matrices, the tetrablock and the pentablock. We also identify the subsets of the bidisc which have the norm-preserving extension property for symmetric functions.