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Carathéodory extremal functions on the symmetrized bidisc

  • Jim Agler
  • Zinaida A. LykovaEmail author
  • N. J. Young
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 271)

Abstract

We show how realization theory can be used to find the solutions of the Carathéodory extremal problem on the symmetrized bidisc
$$ G \,\, \overset{\mathrm{def}}= \{ (z + w, zw): |z| < 1, |w| < 1 \}.$$

We show that, generically, solutions are unique up to composition with automorphisms of the disc. We also obtain formulae for large classes of extremal functions for the Carathéodory problems for tangents of non-generic types.

Keywords

Carathéodory extremal functions symmetrized bidisc model formulae realization formulae 

Mathematics Subject Classification (2010)

32A07 53C22 54C15 47A57 32F45 47A25 30E05 

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at San DiegoSan DiegoUSA
  2. 2.School of Mathematics, Statistics and Physics, Newcastle UniversityNewcastle upon TyneUK
  3. 3.School of Mathematics, Leeds UniversityLeedsUK

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