English

Carath\'eodory extremal functions on the symmetrized bidisc

Complex Variables 2018-05-08 v2

Abstract

We show how realization theory can be used to find the solutions of the Carath\'eodory extremal problem on the symmetrized bidisc G=def{(z+w,zw):z<1,w<1}. G \stackrel{\rm{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 Carath\'eodory problems for tangents of non-generic types.

Keywords

Cite

@article{arxiv.1712.00749,
  title  = {Carath\'eodory extremal functions on the symmetrized bidisc},
  author = {Jim Agler and Zinaida Lykova and N. J. Young},
  journal= {arXiv preprint arXiv:1712.00749},
  year   = {2018}
}

Comments

24 pages, 1 figure. This version contains some minor changes. It is to appear in a volume of Operator Theory: Advamces and Applications, Birkhauser

R2 v1 2026-06-22T23:04:54.267Z