The Journal of Geometric Analysis

, Volume 25, Issue 3, pp 2060–2102 | Cite as

3-Extremal Holomorphic Maps and the Symmetrized Bidisc

Article
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Abstract

We analyze the \(3\)-extremal holomorphic maps from the unit disc \(\mathbb D\) to the symmetrized bidisc \( \mathcal {G}\mathop {=}\limits ^\mathrm{def}\{(z+w,zw): z,w\in \mathbb D\}\) with a view to the complex geometry and function theory of \(\mathcal {G}\). These are the maps whose restriction to any triple of distinct points in \(\mathbb D\) yields interpolation data that are only just solvable. We find a large class of such maps; they are rational of degree at most \(4\). It is shown that there are two qualitatively different classes of rational \(\mathcal {G}\)-inner functions of degree at most \(4\), to be called aligned and caddywhompus functions; the distinction relates to the cyclic ordering of certain associated points on the unit circle. The aligned ones are \(3\)-extremal. We describe a method for the construction of aligned rational \(\mathcal {G}\)-inner functions; with the aid of this method we reduce the solution of a \(3\)-point interpolation problem for aligned holomorphic maps from \(\mathbb D\) to \(\mathcal {G}\) to a collection of classical Nevanlinna–Pick problems with mixed interior and boundary interpolation nodes. Proofs depend on a form of duality for \(\mathcal {G}\).

Keywords

Extremal holomorphic maps Symmetrised bidisc \({\mathcal {G}}\hbox {-inner functions}\) Holomorphic interpolation Invariant distances \(\mu \)-synthesis 

Mathematics Subject Classification

32F45 30E05 93B36 93B50 
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Notes

Acknowledgments

The first author was partially supported by National Science Foundation Grant on Extending Hilbert Space Operators DMS 1068830. The third author was partially supported by the UK Engineering and Physical Sciences Research Council grants EP/J004545/1 and EP/K50340X/1. The collaboration was partially supported by London Mathematical Society Grant 41219.

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Copyright information

© Mathematica Josephina, Inc. 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at San DiegoSan DiegoUSA
  2. 2.School of Mathematics and StatisticsNewcastle UniversityNewcastle upon TyneUK
  3. 3.School of MathematicsLeeds UniversityLeedsUK

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