Nonuniqueness of Carath\'eodory extremal functions on the symmetrized bidisc

TL;DR

This paper explores the nonuniqueness of solutions to the Carathéodory extremal problem on the symmetrized bidisc, providing new insights and descriptions of extremals when multiple solutions exist.

Contribution

It introduces new results characterizing when the Carathéodory extremal problem has multiple solutions on the symmetrized bidisc, extending existing theory with a model formula for Schur class functions.

Findings

Characterization of extremals with unique solutions

Description of multiple extremal solutions

Use of a model formula for Schur class functions

Abstract

We survey the Carath\'eodory extremal problem $\mathrm{Car} \delta$ on the symmetrized bidisc $$ G = \{(z+w,zw):|z|<1, \, |w|<1\} = \{(s,p)\in \mathbb{C}^2: |s-\bar s p| < 1-|p|^2\}. $$ We also give some new results on this topic. We are particularly interested in cases of this problem in which the solution of the problem is not unique. It is known that, for any $\delta=(\lambda,v)\in TG$ with $v\neq 0$, there is at least one $\omega\in\mathbb{T}$ such that $\Phi_\omega$ solves $\mathrm{Car} \delta$, where $\Phi_\omega(s,p) = \frac{2\omega p-s}{2-\omega s}$. Moreover, there is an essentially unique solution of $\mathrm{Car} \delta$ if and only if $\delta$ has exactly one Carath\'eodory extremal function of the form $\Phi_\omega$ for some $\omega\in\mathbb{T}$. We give a description of Carath\'eodory extremals for $\delta\in TG$ with more than one Carath\'eodory extremal function $\Phi_\omega$ for some values of $\omega \in\mathbb{T}$. The proof exploits a model formula for the Schur class of $G$ which is an analog of the well-known network realization formula for Schur-class functions on the disc.