Operator Analysis
Hilbert Space Methods in Complex Analysis
Published on 18. March 2020
394 pages
978-1-108-62120-5 (ISBN)
Description
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-Caratheodory 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.
Reviews / Votes
'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, zbMATHMore details
Series
Language
English
Place of publication
Cambridge
United Kingdom
Target group
College/higher education
Professional and scholarly
Illustrations
Worked examples or Exercises
File size
2,59 MB
ISBN-13
978-1-108-62120-5 (9781108621205)
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Schweitzer Classification
Other editions
Additional editions
Jim Agler | John Edward McCarthy | Nicholas John Young
Operator Analysis
Hilbert Space Methods in Complex Analysis
Book
03/2020
Cambridge University Press
€165.50
Shipment within 15-20 days
Persons
Content
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. Caratheodory-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.
