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Geodesics Retracts and the Norm Preserving Extension Property in the Symmetrized Bidisc 1st Edition by Jim Agler, Zinaida Lykova, Nicholas Young ISBN 9781470450755 1470450755

Name: Geodesics Retracts and the Norm Preserving Extension Property in the Symmetrized Bidisc 1st Edition by Jim Agler, Zinaida Lykova, Nicholas Young ISBN 9781470450755 1470450755
SKU: EB-51652350
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Geodesics Retracts and the Norm Preserving Extension Property in the Symmetrized Bidisc 1st Edition Jim Agler Digital Instant Download

Author(s): Jim Agler; Zinaida Lykova; Nicholas Young

ISBN(s): 9781470450755, 1470450755

Edition: 1

File Details: PDF, 1012 KB

Year: 2019

Language: English

SKU: EB-51652350 Category: Uncategorized Tags: Jim Agler, Nicholas Young, Zinaida Lykova

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Geodesics Retracts and the Norm Preserving Extension Property in the Symmetrized Bidisc 1st Edition by Jim Agler, Zinaida Lykova, Nicholas Young – Ebook PDF Instant Download/Delivery: 9781470450755 ,1470450755
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Product details:
ISBN 10: 1470450755
ISBN 13: 9781470450755
Author: Jim Agler, Zinaida Lykova, Nicholas Young

A set V in a domain U in Cn 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 Gdef = {(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, z2 ) : |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 Γ-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 ×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.

Geodesics Retracts and the Norm Preserving Extension Property in the Symmetrized Bidisc 1st Edition Table of contents:

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 $G$ -inner Functions

Chapter 5. The Retracts of $G$ and the Bidisc $D2mathbb{D}^2$
5.1. Retracts and Geodesics of $G$
5.2. Retracts of $D2mathbb{D}^2$
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 $R \cup D$ with the Norm-Preserving Extension Property in $G$
10.1. Definitions and Lemmas
10.2. The Proof of the Norm-Preserving Extension Property for $R \cup D$

Chapter 11. $V$ and a Circular Region $R$ in the Plane

Chapter 12. Proof of the Main Theorem
12.1. Preliminary Lemmas
12.2. $σ : R \to V$ is Analytic on $R^-$
12.3. Degree Considerations

Chapter 13. Sets in $D2mathbb{D}^2$ 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 $ΦωPhi_{omega}$ are Extremal

Appendix B. Types of Geodesic: A Crib and Some Cartoons
B.1. Crib Sheet
B.2. Cartoons

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Tags: Jim Agler, Zinaida Lykova, Nicholas Young, Geodesics Retracts, Norm Preserving, Extension Property, Symmetrized Bidisc

Geodesics Retracts and the Norm Preserving Extension Property in the Symmetrized Bidisc 1st Edition by Jim Agler, Zinaida Lykova, Nicholas Young ISBN 9781470450755 1470450755

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