Nonuniqueness of Carathéodory Extremal Functions on the Symmetrized Bidisc

We survey the Carathéodory extremal problem Car δ on symmetrized bidisc $$G\,\,\mathop = \limits^{{\rm{def}}} \,\,\left\{ {\left( {z + w,zw} \right):\left| z \right| < 1,\,\,\,\left| w 1} \right\} \left\{ {s,p} \right) \in {\mathbb{C}^2}:\left| {s - \bar sp} 1 {{\left| p \right|}^2}} \right\}.$$ also give some new results this topic. are particularly interested in cases of which solution is not unique. It known that, for any (λ, v) ∈ TG with v ≠ 0, there at least one $$\omega \mathbb{T}$$ such that Φω solves δ, where $${\Phi _\omega }\left( {{2\omega s} \over {2 \omega s}}$$ . Moreover, an essentially unique if and only has exactly function form a description extremals more than values The proof exploits model formula Schur class G ana log well-known network realization Schur-class functions disc.