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Operator Analysis
Hilbert Space Methods in Complex Analysis


Part of Cambridge Tracts in Mathematics

  • Date Published: March 2020
  • availability: Available
  • format: Hardback
  • isbn: 9781108485449

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About the Authors
  • 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
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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

  • Authors

    Jim Agler, University of California, San Diego
    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, Washington University, St Louis
    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, University of Leeds and University of Newcastle
    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.

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