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E-raamat: Operator Analysis: Hilbert Space Methods in Complex Analysis

(University of California, San Diego), (Washington University, St Louis),
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Operator Analysis: Hilbert Space Methods in Complex AnalysisSuurem pilt
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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.

Arvustused

'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

Muu info

A detailed monograph exploring how operator theory interacts with function theory in one and several variables.
Preface xiii
Acknowledgments xv
PART ONE COMMUTATIVE THEORY
1(2)
1 The Origins of Operator-Theoretic Approaches to Function Theory
3(1)
1.1 Operators
3(1)
1.2 Functional Calculi
4(5)
1.3 Operators on Hilbert Space
9(3)
1.4 The Spectral Theorem
12(1)
1.5 Hardy Space and the Unilateral Shift
13(2)
1.6 Invariant Subspaces of the Unilateral Shift'
15(1)
1.7 Von Neumann's Theory of Spectral Sets
16(3)
1.8 The Schur Class and Spectral Domains
19(2)
1.9 The Sz.-Nagy Dilation Theorem
21(1)
1.10 Ando's Dilation Theorem
22(1)
1.11 The Sz.-Nagy--Foias Model Theory
23(2)
1.12 The Sarason Interpolation Theorem
25(5)
1.13 Historical Notes
30(3)
2 Operator Analysis on D: Model Formulas, Lurking Isometries, and Positivity Arguments
33(1)
2.1 Overview
33(1)
2.2 A Model Formula for (D)
33(4)
2.3 Reproducing Kernel Hilbert Spaces
37(2)
2.4 Lurking Isometries
39(5)
2.5 The Network Realization Formula (Scalar Case) via Model Theory
44(8)
2.6 Interpolation via Model Theory
52(4)
2.7 The Muntz--Szasz Interpolation Theorem
56(5)
2.8 Positivity Arguments
61(9)
2.9 Historical Notes
70(1)
3 Further Development of Models on the Disc
71(1)
3.1 A Model Formula for LB(H,K)(D)
71(3)
3.2 Lurking Isometries Revisited
74(1)
3.3 The Network Realization Formula
75(3)
3.4 Tensor Products of Hilbert Spaces
78(1)
3.5 Tensor Products of Operators
79(2)
3.6 Realization of Rational Matrix Functions and the McMillan Degree
81(2)
3.7 Pick Interpolation Revisited
83(2)
3.8 The Corona Problem
85(6)
3.9 Historical Notes
91(2)
4 Operator Analysis on D2
93(1)
4.1 The Space of Hereditary Functions on D2
93(2)
4.2 The Hereditary Functional Calculus on D2
95(4)
4.3 Models on D2
99(5)
4.4 Models on D2 via the Duality Construction
104(2)
4.5 The Network Realization Formula for D2
106(3)
4.6 Nevanlinna--Pick Interpolation on D2
109(3)
4.7 Toeplitz Corona for the Bidisc
112(1)
4.8 Operator-Valued Functions on D2
113(3)
4.9 Models of Operator-Valued Functions on D2
116(13)
4.10 Historical Notes
129(2)
5 Caratheodory--Julia Theory on the Disc and the Bidisc
131(1)
5.1 The One-Variable Results
131(2)
5.2 The Model Approach to Regularity on D: B-points and C-points
133(4)
5.3 A Proof of the Caratheodory--Julia Theorem on D via Models
137(4)
5.4 Pick Interpolation on the Boundary
141(2)
5.5 Regularity, B-points and C-points on the Bidisc
143(3)
5.6 The Missing Link
146(1)
5.7 Historical Notes
147(1)
6 Herglotz and Nevanlinna Representations in Several Variables
148(1)
6.1 Overview
148(1)
6.2 The Herglotz Representation on B2
149(4)
6.3 Nevanlinna Representations on H via Operator Theory
153(3)
6.4 The Nevanlinna Representations on Hf2
156(4)
6.5 A Classification Scheme for Nevanlinna Representations in Two Variables
160(5)
6.6 The Type of a Function
165(3)
6.7 Historical Notes
168(1)
7 Model Theory on the Symmetrized Bidisc
169(1)
7.1 Adding Symmetry to the Fundamental Theorem for D2
170(3)
7.2 How to Define Models on the Symmetrized Bidisc
173(3)
7.3 The Network Realization Formula for G
176(1)
7.4 The Hereditary Functional Calculus on G
177(5)
7.5 When Is G a Spectral Domain?
182(3)
7.6 G Spectral Implies G Complete Spectral
185(1)
7.7 The Spectral Nevanlinna--Pick Problem
185(2)
7.8 Historical Notes
187(2)
8 Spectral Sets: Three Case Studies
189(1)
8.1 Von Neumann's Inequality and the Pseudo-Hyperbolic Metric on D
189(3)
8.2 Spectral Domains and the Caratheodory Metric
192(2)
8.3 Background Material
194(1)
8.4 Lempert's Theorem
195(8)
8.5 The Caratheodory Distance on G
203(2)
8.6 Von Neumann's Inequality on Subvarieties of the Bidisc
205(6)
8.7 Historical Notes
211(2)
9 Calcular Norms
213(1)
9.1 The Taylor Spectrum and Functional Calculus
213(2)
9.2 Calcular Norms and Algebras
215(8)
9.3 Halmos's Conjecture and Paulsen's Theorem
223(3)
9.4 The Douglas-Paulsen Norm
226(7)
9.5 The B. and F. Delyon Norm and Crouzeix's Theorem
233(2)
9.6 The Badea--Beckermann--Crouzeix Norm
235(8)
9.7 The Polydisc Norm
243(2)
9.8 The Oka Extension Theorem and Calcular Norms
245(7)
9.9 Historical Notes
252(2)
10 Operator Monotone Functions
254(1)
10.1 Lowner's Theorems
254(8)
10.2 An Interlude on Linear Programming
262(7)
10.3 Locally Matrix Monotone Functions in d Variables
269(9)
10.4 The Lowner Class in d Variables
278(1)
10.5 Globally Monotone Rational Functions in Two Variables
279(3)
10.6 Historical Notes
282(1)
PART TWO NON-COMMUTATIVE THEORY
283(2)
11 Motivation for Non-Commutative Functions
285(1)
11.1 Non-Commutative Polynomials
285(2)
11.2 Sums of Square
287(1)
11.3 Nullstellensatz
287(1)
11.4 Linear Matrix Inequalities
288(1)
11.5 The Implicit Function Theorem
289(1)
11.6 Matrix Monotone Functions
289(1)
11.7 The Functional Calculus
290(1)
11.8 Historical Notes
290(1)
12 Basic Properties of Non-Commutative Functions
291(1)
12.1 Definition of an nc-Function
291(4)
12.2 Locally Bounded nc-Functions Are Holomorphic
295(3)
12.3 Nc Topologies
298(2)
12.4 Historical Notes
300(2)
13 Montel Theorems
302(1)
13.1 Overview
302(1)
13.2 Wandering Isometries
303(2)
13.3 A Graded Montel Theorem
305(4)
13.4 An nc Montel Theorem
309(1)
13.5 Closed Cones
310(2)
13.6 Historical Notes
312(1)
14 Free Holomorphic Functions
313(1)
14.1 The Range of a Free, Holomorphic Function
313(2)
14.2 Nc-Models and Free Realizations
315(4)
14.3 Free Pick Interpolation
319(5)
14.4 Free Realizations Redux
324(2)
14.5 Extending off Varieties
326(1)
14.6 The nc Oka--Weil Theorem
327(1)
14.7 Additional Results
328(1)
14.8 Historical Notes
329(1)
15 The Implicit Function Theorem
330(1)
15.1 The Fine Inverse Function Theorem
330(2)
15.2 The Fat Inverse Function Theorem
332(5)
15.3 The Implicit Function Theorem
337(2)
15.4 The Range of an nc-Function
339(2)
15.5 Applications of the Implicit Function Theorem in Non-Commutative Algebraic Geometry
341(2)
15.6 Additional Results
343(1)
15.7 Historical Notes
343(1)
16 Non-Commutative Functional Calculus
344(1)
16.1 Nc Operator Functions
344(5)
16.2 Polynomial Approximation and Power Series
349(5)
16.3 Extending Free Functions to Operators
354(2)
16.4 Nc Functional Calculus in Banach Algebras
356(1)
16.5 Historical Notes
357(1)
Notation 358(3)
Bibliography 361(11)
Index 372
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.
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