Nonuniqueness of Carathéodory extremal functions on the symmetrized bidisc

We survey the Carathéodory extremal problem $\mathrm{Car} δ$ 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 $δ=(λ,v)\in TG$ with $v\neq 0$, there is at least one $ω\in\mathbb{T}$ such that $Φ_ω$ solves $\mathrm{Car} δ$, where $Φ_ω(s,p) = \frac{2ωp-s}{2-ωs}$. Moreover, there is an essentially unique solution of $\mathrm{Car} δ$ if and only if $δ$ has exactly one Carathéodory extremal function of the form $Φ_ω$ for some $ω\in\mathbb{T}$. We give a description of Carathéodory extremals for $δ\in TG$ with more than one Carathéodory extremal function $Φ_ω$ for some values of $ω\in\mathbb{T}$. 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.

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