Operator Analysis
Operator Analysis
John Edward McCarthy, Washington University, St Louis
Nicholas John Young, University of Leeds and University of Newcastle
This book shows how operator theory interacts with function theory in one and several variables. The authors develop the theory in detail, leading the reader to the cutting edge of contemporary research. It starts with a treatment of the theory of bounded holomorphic functions on the unit disc. Model theory and the network realization formula are used to solve Nevanlinna-Pick interpolation problems, and the same techniques are shown to work on the bidisc, the symmetrized bidisc, and other domains. The techniques are powerful enough to prove the Julia-Carathéodory theorem on the bidisc, Lempert's theorem on invariant metrics in convex domains, the Oka extension theorem, and to generalize Loewner's matrix monotonicity results to several variables. In Part II, the book gives an introduction to non-commutative function theory, and shows how model theory and the network realization formula can be used to understand functions of non-commuting matrices.
- Shows how function theory on the unit disc, properly formulated, can transfer to the bidisc, and other domains in several complex variables
- Illustrates the power of network realization formulas, even in the non-commutative setting
- Provides a self-contained introduction to non-commutative function theory
Reviews & endorsements
'This is a much awaited book, which brings together several results obtained in the last decades, pertaining to the applications of operator theory in Hilbert space to function theory … The book is extremely nicely written. It does not need many prerequisites, besides elementary facts of complex analysis and functional analysis; and it can be of much use to interested researchers as well as to graduate students.' Dan Timotin, zbMATH
Product details
No date availableAdobe eBook Reader
9781108621205
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Table of Contents
- Part I. Commutative Theory:
- 1. The origins of operator-theoretic approaches to function theory
- 2. Operator analysis on D: model formulas, lurking Isometries, and positivity arguments
- 3. Further development of models on the disc
- 4. Operator analysis on D2
- 5. Carathéodory-Julia theory on the disc and the bidisc
- 6. Herglotz and Nevanlinna representations in several variables
- 7. Model theory on the symmetrized bidisc
- 8. Spectral sets: three case studies
- 9. Calcular norms
- 10. Operator monotone functions
- Part II. Non-Commutative Theory:
- 11. Motivation for non-commutative functions
- 12. Basic properties of non-commutative functions
- 13. Montel theorems
- 14. Free holomorphic functions
- 15. The implicit function theorem
- 16. Noncommutative functional calculus
- Notation.
