Nonuniqueness of Carathéodory extremal functions on the symmetrized bidisc

Abstract

We survey the Carathéodory 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 $v\ne 0$, 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 Carathéodory extremal function of the form ${\Phi }_{\omega }$ for some $\omega \in \mathbb{T}$. We give a description of Carathéodory extremals for $\delta \in TG$ with more than one Carathéodory 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.

18 pages. The paper was accepted on 12th January 2022 to appear in Analysis Mathematica. arXiv admin note: text overlap with arXiv:1712.00749