Operator Analysis
Hilbert Space Methods in Complex Analysis
Part of Cambridge Tracts in Mathematics
- Authors:
- Jim Agler, University of California, San Diego
- John Edward McCarthy, Washington University, St Louis
- Nicholas John Young, University of Leeds and University of Newcastle
- Date Published: March 2020
- availability: Available
- format: Hardback
- isbn: 9781108485449
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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.
Read more- 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
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×Product details
- Date Published: March 2020
- format: Hardback
- isbn: 9781108485449
- length: 388 pages
- dimensions: 235 x 157 x 25 mm
- weight: 0.66kg
- availability: Available
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
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