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Geodesics, Retracts, and the Norm-Preserving Extension Property in the Symmetrized Bidisc

About this Title

Jim Agler, Department of Mathematics, University of California at San Diego, California 92103, Zinaida Lykova, School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom and Nicholas Young, School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom — and — School of Mathematics, Leeds University, Leeds LS2 9JT, United Kingdom

Publication: Memoirs of the American Mathematical Society
Publication Year: 2019; Volume 258, Number 1242
ISBNs: 978-1-4704-3549-3 (print); 978-1-4704-5075-5 (online)
DOI: https://doi.org/10.1090/memo/1242
Published electronically: February 22, 2019
Keywords: Symmetrized bidisc, complex geodesic, norm-preserving extension property, holomorphic retract, spectral set, Kobayashi extremal problem, Carathéodory extremal problem, von Neumann inequality, semialgebraic set
MSC: Primary 32A07, 53C22, 54C15, 47A57, 32F45; Secondary 47A25, 30E05

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Table of Contents

Chapters

  • Preface
  • 1. Introduction
  • 2. An overview
  • 3. Extremal problems in the symmetrized bidisc $G$
  • 4. Complex geodesics in $G$
  • 5. The retracts of $G$ and the bidisc $\mathbb {D}^2$
  • 6. Purely unbalanced and exceptional datums in $G$
  • 7. A geometric classification of geodesics in $G$
  • 8. Balanced geodesics in $G$
  • 9. Geodesics and sets $V$ with the norm-preserving extension property in $G$
  • 10. Anomalous sets $\mathcal {R}\cup \mathcal {D}$ with the norm-preserving extension property in $G$
  • 11. $V$ and a circular region $R$ in the plane
  • 12. Proof of the main theorem
  • 13. Sets in $\mathbb {D}^2$ with the symmetric extension property
  • 14. Applications to the theory of spectral sets
  • 15. Anomalous sets with the norm-preserving extension property in some other domains
  • A. Some useful facts about the symmetrized bidisc
  • B. Types of geodesic: a crib and some cartoons

Abstract

A set $V$ in a domain $U$ in $\mathbb {C}^n$ has the norm-preserving extension property if every bounded holomorphic function on $V$ has a holomorphic extension to $U$ with the same supremum norm. We prove that an algebraic subset of the symmetrized bidisc \[ G \eqdef{(z+w,zw):|z|<1, |w| < 1} \] has the norm-preserving extension property if and only if it is either a singleton, $G$ itself, a complex geodesic of $G$, or the union of the set $\{(2z,z^2): |z|<1\}$ and a complex geodesic of degree $1$ in $G$. We also prove that the complex geodesics in $G$ coincide with the nontrivial holomorphic retracts in $G$. Thus, in contrast to the case of the ball or the bidisc, there are sets in $G$ which have the norm-preserving extension property but are not holomorphic retracts of $G$. In the course of the proof we obtain a detailed classification of the complex geodesics in $G$ modulo automorphisms of $G$. We give applications to von Neumann-type inequalities for $\Gamma$-contractions (that is, commuting pairs of operators for which the closure of $G$ is a spectral set) and for symmetric functions of commuting pairs of contractive operators. We find three other domains that contain sets with the norm-preserving extension property which are not retracts: they are the spectral ball of $2\times 2$ matrices, the tetrablock and the pentablock. We also identify the subsets of the bidisc which have the norm-preserving extension property for symmetric functions.

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