Noncommutative Function-Theoretic Operator Theory and Applications
Noncommutative Function-Theoretic Operator Theory and Applications
$146.99
USDThis concise monograph explores how core ideas in Hardy space function theory and operator theory continue to be useful and informative in new settings, leading to new insights for noncommutative multivariable operator theory. Beginning with a review of the confluence of system theory ideas and reproducing kernel techniques, the book then covers representations of backward-shift-invariant subspaces in the Hardy space as ranges of observability operators, and representations for forward-shift-invariant subspaces via a Beurling–Lax representer equal to the transfer function of the linear system. This pair of backward-shift-invariant and forward-shift-invariant subspace form a generalized orthogonal decomposition of the ambient Hardy space. All this leads to the de Branges–Rovnyak model theory and characteristic operator function for a Hilbert space contraction operator. The chapters that follow generalize the system theory and reproducing kernel techniques to enable an extension of the ideas above to weighted Bergman space multivariable settings.
- Presents a broad overview of the whole book in the context of the classical Hardy space function theory setting, providing readers with an easy reference point for comparing where the general setting still carries the classical structure, and where and how it departs significantly from the classical setting
- Includes remarks and examples throughout the text, helping readers understand the broader context of the material, how a result foreshadows a more general result to come, and how loose ends in a current result are resolved by a more sophisticated result to come in a later chapter
- Provides a Notes section at the end of each chapter to point readers to recent literature with results closely related to those in the book and possible new directions for future research
- Makes explicit connections with the work on Bergman-inner functions and the Bergman shift operator from the 1990s
Reviews & endorsements
‘Noncommutative Function-Theoretic Operator Theory and Applications by Ball and Bolotnikov is a comprehensive monograph by acknowledged experts in the fields of operator theory and function theory. It gives an account of a very active area of modern research, to which the authors themselves have been major contributors. The significant themes of the book include reproducing kernel Hilbert spaces (notably weighted Bergman spaces), Beurling-Lax theorems, and systems-theoretic ideas expressed in operator-theoretic terms. The work as a whole is presented in a multivariable noncommutative context, and thus extends classical work on Hardy-space function theory and related operator theory.' Jonathan Partington, University of Leeds
'This book is a comprehensive treatise on a fascinating area of mathematical analysis by two prominent scholars in the field, who have themselves made many original contributions to the theory and are renowned for their knowledge of the research literature. The authors demonstrate that much of the theories of operators on Hilbert space and of the theory of functions of one or more complex variables can be generalized to a much broader setting. The theories are also applied to functions of non-commuting variables, a topic which has come under intensive study by many specialists only in the last few decades. A principal theme of the book is 'systems theory', which, despite its origins in engineering, is here generalized and shown to be a powerful tool for the analysis of functions of non-commuting variables and of non-commuting tuples of operators. I recommend this tremendous resource strongly to the operator theory community.' Nicholas Young, University of Leeds and Newcastle University
Product details
December 2021Hardback
9781316518991
300 pages
235 × 159 × 32 mm
0.813kg
Available
Table of Contents
- 1. Introduction
- 2. Formal Reproducing Kenel Hilbert Spaces
- 3. Contractive multipliers
- 4. Stein relations and observability range spaces
- 5. Beurling-Lax theorems based on contractive multipliers
- 6. Non-orthogonal Beurling-Lax representations
- 7. Orthogonal Beurling-Lax representations
- 8. Models for ω-hypercontractive operator tuples
- 9. Regular formal power series.
