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

DOI10.1007/S10476-022-0138-6arXiv2201.09078MaRDI QIDQ6388888

Zinaida A. Lykova, N. J. Young, Jim Agler

Publication date: 22 January 2022

Abstract: We survey the Carath'eodory extremal problem $mathrmCardelta$ on the symmetrized bidisc ${\displaystyle }$ G = {(z+w,zw):|z|<1, , |w|<1} = {(s,p)in mathbb{C}^2: |s-�ar s p| < 1-|p|^2}. ${\displaystyle }$ 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 $delta=(lambda,v)inTG$ with $veq0$, there is at least one $omegainmathbbT$ such that $Ph{i}_{o}mega$ solves $mathrmCardelta$, where $Ph{i}_{o}mega(s,p)=frac2omegap-s2-omegas$. Moreover, there is an essentially unique solution of $mathrmCardelta$ if and only if $delta$ has exactly one Carath'eodory extremal function of the form $Ph{i}_{o}mega$ for some $omegainmathbbT$. We give a description of Carath'eodory extremals for $deltainTG$ with more than one Carath'eodory extremal function $Ph{i}_{o}mega$ for some values of $omegainmathbbT$. 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.

Mathematics Subject Classification ID

Invariant metrics and pseudodistances in several complex variables (32F45) Linear operator methods in interpolation, moment and extension problems (47A57) Geodesics in global differential geometry (53C22) Moment problems and interpolation problems in the complex plane (30E05) Retraction (54C15) Spectral sets of linear operators (47A25) Holomorphic functions of several complex variables (32A99)

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