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

Cover -- Title page -- Preface -- Chapter 1. Introduction -- Chapter 2. An overview -- Chapter 3. Extremal problems in the symmetrized bidisc -- 3.1. The Carathéodory and Kobayashi extremal problems -- 3.2. The Carathéodory extremal problem ( ) for -- 3.3. Five types of datum in -- 3.4. The Kobayashi extremal problem ( ) for -- Chapter 4. Complex geodesics in -- 4.1. Complex geodesics and datums in -- 4.2. Uniqueness of complex geodesics for each datum in -- 4.3. Flat C-geodesics -- 4.4. Rational \Ga-inner functions -- Chapter 5. The retracts of and the bidisc ² -- 5.1. Retracts and geodesics of -- 5.2. Retracts of ² -- 5.3. Geodesics in are varieties -- Chapter 6. Purely unbalanced and exceptional datums in -- Chapter 7. A geometric classification of geodesics in -- Chapter 8. Balanced geodesics in -- Chapter 9. Geodesics and sets with the norm-preserving extension property in -- 9.1. and ( ) -- 9.2. and balanced datums -- 9.3. and flat or royal datums -- Chapter 10. Anomalous sets ℛ∪ with the norm-preserving extension property in -- 10.1. Definitions and lemmas -- 10.2. The proof of the norm-preserving extension property for \calr∪\cald -- Chapter 11. and a circular region in the plane -- Chapter 12. Proof of the main theorem -- 12.1. Preliminary lemmas -- 12.2. : → is analytic on ⁻ -- 12.3. Degree considerations -- Chapter 13. Sets in 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 and Γ -- A.2. Complex C-geodesics in -- A.3. Automorphisms of -- A.4. A trichotomy theorem -- A.5. Datums for which all Φ_{ } are extremal -- Appendix B. Types of geodesic: a crib and some cartoons.

B.1. Crib sheet -- B.2. Cartoons -- Bibliography -- Index -- Back Cover.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.