Operator Analysis: Hilbert Space Methods in Complex Analysis

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Jim Agler (University of California, San Diego) John Edward McCarthy (Washington University, St Louis)

Operator Analysis

Hilbert Space Methods in Complex Analysis

Jim Agler (University of California, San Diego), John Edward McCarthy (Washington University, St Louis), Nicholas John Young

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English

Cambridge University Press
26 March 2020

Complex analysis, complex variables; Functional analysis & transforms

Series: Cambridge Tracts in Mathematics

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.

By: Jim Agler (University of California San Diego), John Edward McCarthy (Washington University, St Louis), Nicholas John Young
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Dimensions: Height: 235mm, Width: 157mm, Spine: 25mm
Weight: 660g
ISBN: 9781108485449
ISBN 10: 1108485448
Series: Cambridge Tracts in Mathematics
Pages: 388
Publication Date: 26 March 2020
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

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.

Jim Agler is Distinguished Professor Emeritus at the University of California, San Diego. He received the G. de B. Robinson award from the Canadian Mathematical Society in 2016 and delivered the 2017 London Mathematical Society Invited Lectures. He is the co-author of Pick Interpolation and Hilbert Function Spaces (2002). John Edward McCarthy is the Spencer T. Olin Professor of Arts and Sciences at Washington University, St Louis, and chair of the Department of Mathematics and Statistics. He received the G. de B. Robinson award from the Canadian Mathematical Society (2016) and was co-author of Pick Interpolation and Hilbert Function Spaces (2002). Nicholas John Young is Research Professor at Leeds University and Senior Research Investigator at University of Newcastle upon Tyne. He is the author of An Introduction to Hilbert Space (Cambridge, 1988) and approximately 100 research articles in analysis.

Reviews for Operator Analysis: Hilbert Space Methods in Complex Analysis

'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