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

Abstract

Preface vii Chapter 1. Introduction Chapter 2. An overview Chapter 3. Extremal problems in the symmetrized bidisc G 3.1. The Carathéodory and Kobayashi extremal problems 3.2. The Carathéodory extremal problem Car(δ) for G 3.3. Five types of datum δ in G 3.4. The Kobayashi extremal problem Kob(δ) for G Chapter 4. Complex geodesics in G 4.1. Complex geodesics and datums in G 4.2. Uniqueness of complex geodesics for each datum in G 4.3. Flat C-geodesics 4.4. Rational Γ-inner functions Chapter 5. The retracts of G and the bidisc D 2 5.1. Retracts and geodesics of G 5.2. Retracts of D 2 5.3. Geodesics in G are varieties Chapter 6. Purely unbalanced and exceptional datums in G Chapter 7. A geometric classification of geodesics in G Chapter 8. Balanced geodesics in G Chapter 9. Geodesics and sets V with the norm-preserving extension property in G 9.1. V and Car(δ) 9.2. V and balanced datums 9.3. V and flat or royal datums Chapter 10. Anomalous sets R ∪ D with the norm-preserving extension property in G 10.1. Definitions and lemmas 10.2. The proof of the norm-preserving extension property for R ∪ D Chapter 11. V and a circular region R in the plane Chapter 12. Proof of the main theorem iii iv CONTENTS 12.1. Preliminary lemmas 12.2. σ : R → V is analytic on R − 12.3. Degree considerations Chapter 13. Sets in D 2 with the symmetric extension property Chapter 14. Applications to the theory of spectral sets Chapter 15. Anomalous sets with the norm-preserving extension property in some other domains Appendix A. Some useful facts about the symmetrized bidisc A.1. Basic properties of G and Γ A.2. Complex C-geodesics in G A.3. Automorphisms of G A.4. A trichotomy theorem A.5. Datums for which all Φ ω are extremal Appendix B. Types of geodesic: a crib and some cartoons B.1. Crib sheet B.2. Cartoons Bibliography Index

Carathéodory extremal functions on the symmetrized bidisc

“…These types correspond to our terminology from [2] (or from Section 1) in the following way. Recall that an automorphism of D is either the identity, elliptic, parabolic or hyperbolic, meaning that the set {z ∈ D − : m(z) = z} consists of either all of D − , a single point of D, a single point of T or two points in T.…”

“…The last inequality distinguishes purely unbalanced from exceptional tangents -the left hand side of equation (1.4) is equal to zero for exceptional tangents. The five types of tangent are discussed at length in our paper [2]. We proved [2, Theorem 3.6] a 'pentachotomy theorem', which states that every nondegenerate tangent in T G is of exactly one of the above five types.…”

“…Thus the class of all well-aligned Carathéodory extremal functions for δ is given by the set of norm-preserving analytic extensions to G of g in equation (4.3), as h ranges over functions in the Schur class taking the value m(η) at ζ. Typically there will be many such extensions of g, as can be seen from the proof of [2,Theorem 10.1]. An extension is obtained as the Cayley transform of a function defined by a Herglotz-type integral with respect to a probability measure µ on T 2 .…”

“…An extension is obtained as the Cayley transform of a function defined by a Herglotz-type integral with respect to a probability measure µ on T 2 . In the proof of [2,Lemma 10.8], µ is chosen to be the product of two measures µ R and µ F on T; examination of the proof shows that one can equally well choose any measure µ on T 2 such that…”

“…In this paper we continue along the Bart-Gohberg-Kaashoek path by using realization theory to prove results in complex geometry. Specifically, we are interested in the geometry of the symmetrized bidisc a domain in C 2 that has been much studied in the last two decades: see [8,9,11,7,17,18,2], along with many other papers. We shall use realization theory to prove detailed results about the Carathéodory extremal problem on G, defined as follows (see [14,12]).…”