Pick Interpolation And Hilbert Function Spaces
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Pick Interpolation And Hilbert Function Spaces
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Author : Jim Agler
language : en
Publisher: American Mathematical Society
Release Date : 2023-02-22
Pick Interpolation And Hilbert Function Spaces written by Jim Agler and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-02-22 with Mathematics categories.
The book first rigorously develops the theory of reproducing kernel Hilbert spaces. The authors then discuss the Pick problem of finding the function of smallest $H^infty$ norm that has specified values at a finite number of points in the disk. Their viewpoint is to consider $H^infty$ as the multiplier algebra of the Hardy space and to use Hilbert space techniques to solve the problem. This approach generalizes to a wide collection of spaces. The authors then consider the interpolation problem in the space of bounded analytic functions on the bidisk and give a complete description of the solution. They then consider very general interpolation problems. The book includes developments of all the theory that is needed, including operator model theory, the Arveson extension theorem, and the hereditary functional calculus.
Operator Analysis
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Author : Jim Agler
language : en
Publisher: Cambridge University Press
Release Date : 2020-03-26
Operator Analysis written by Jim Agler and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-03-26 with Mathematics categories.
This monograph, aimed at graduate students and researchers, explores the use of Hilbert space methods in function theory. Explaining how operator theory interacts with function theory in one and several variables, the authors journey from an accessible explanation of the techniques to their uses in cutting edge research.
Nevanlinna Pick Spaces And Dilations
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Author : Michael Peter Hartz
language : en
Publisher:
Release Date : 2016
Nevanlinna Pick Spaces And Dilations written by Michael Peter Hartz and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016 with Dilation theory (Operator theory) categories.
The majority of this thesis is devoted to the study of Nevanlinna-Pick spaces and their multiplier algebras. These spaces are Hilbert function spaces in which a version of the Nevanlinna-Pick interpolation theorem from complex analysis holds. Their multiplier algebras occupy an important place at the interface between operator algebras, operator theory and complex analysis. Over the last few years, the classification problem for these algebras has attracted considerable attention. These investigations were pioneered by Davidson, Ramsey and Shalit, who used a theorem of Agler and McCarthy to identify a given multiplier algebra with the restriction of the multiplier algebra of the universal Nevanlinna-Pick space, namely the Drury-Arveson space, to an analytic variety in a complex ball. In this thesis, the classification problem is studied from three different angles. In Chapter 3, we investigate multiplier algebras associated to embedded discs in a complex ball. In particular, we exhibit uncountably many embedded discs which are biholomorphic in a strong sense, but whose multiplier algebras are not isomorphic. Motivated by these issues, we use in Chapter 4 a different approach to the classification problem. Thus, we study the spaces and their multiplier algebras directly without making use of the existence of a universal Nevanlinna-Pick space. This allows us to completely classify the multiplier algebras of a special class of spaces on homogeneous varieties. In Chapter 5, we investigate the complexity of this classification problem from the point of view of Borel complexity theory. In Chapter 6, we show that the Hardy space on the unit disc is essentially the only Nevanlinna-Pick space whose multiplication operators are all hyponormal. The last part of this thesis is concerned with dilations and von Neumann's inequality. It has been known since the seventies that there are three commuting contractions which do not satisfy von Neumann's inequality. In Chapter 7, we show that every tuple of commuting contractions which forms a multivariable weighted shift dilates to a tuple of commuting unitaries and hence satisfies von Neumann's inequality, thereby providing a positive answer to a question of Shields and Lubin from 1974.
Interpolation And Sampling In Spaces Of Analytic Functions
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Author : Kristian Seip
language : en
Publisher: American Mathematical Soc.
Release Date : 2004
Interpolation And Sampling In Spaces Of Analytic Functions written by Kristian Seip and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Mathematics categories.
Based on a series of six lectures given by the author at the University of Michigan, this book is intended as an introduction to the topic of interpolation and sampling in analytic function spaces. The three major topics covered are Nevanlinna-Pick interpolation, Carleson's interpolation theorem, an
Topics In Interpolation Theory
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Author : Bernd Fritzsche
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06
Topics In Interpolation Theory written by Bernd Fritzsche and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
About one half of the papers in this volume are based on lectures which were pre sented at a conference at Leipzig University in August 1994, which was dedicated to Vladimir Petrovich Potapov. He would have been eighty years old. These have been supplemented by: (1) Historical material, based on reminiscences of former colleagues, students and associates of V.P. Potapov. (2) Translations of a number of important papers (which serve to clarify the Potapov approach to problems of interpolation and extension, as well as a number of related problems and methods) and are relatively unknown in the West. (3) Two expository papers, which have been especially written for this volume. For purposes of discussion, it is convenient to group the technical papers in this volume into six categories. We will now run through them lightly, first listing the major theme, then in parentheses the authors of the relevant papers, followed by discussion. Some supplementary references are listed at the end; OT72 which appears frequently in this volume, refers to Volume 72 in the series Operator Theory: Advances and Applications. It was dedicated to V.P. Potapov. 1. Multiplicative decompositions (Yu.P. Ginzburg; M.S. Livsic, I.V. Mikhailova; V.I. Smirnov).
Pick Interpolation And The Distance Formula
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Author : Ryan Hamilton
language : en
Publisher:
Release Date : 2012
Pick Interpolation And The Distance Formula written by Ryan Hamilton and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with categories.
The classical interpolation theorem for the open complex unit disk, due to Nevanlinna and Pick in the early 20th century, gives an elegant criterion for the solvability of the problem as an eigenvalue problem. In the 1960s, Sarason reformulated problems of this type firmly in the language of operator theoretic function theory. This thesis will explore connections between interpolation problems on various domains (both single and several complex variables) with the viewpoint that Sarason's work suggests. In Chapter 1, some essential preliminaries on bounded operators on Hilbert space and the functionals that act on them will be presented, with an eye on the various ways distances can be computed between operators and a certain type of ideal. The various topologies one may define on B(H) will play a prominent role in this development. Chapter 2 will introduce the concept of a reproducing kernel Hilbert space, and a distinguished operator algebra that we associate to such spaces known as the multiplier algebra. The various operator theoretic properties that multiplier algebras enjoy will be presented, with a particular emphasis on their invariant subspace lattices and the connection to distance formulae. In Chapter 3, the Nevanlinna-Pick problem will be invested in general for any reproducing kernel Hilbert space, with the basic heuristic for distance formulae being presented. Chapter 4 will treat a large class of reproducing kernel Hilbert spaces associated to measure spaces, where a Pick-like theorem will be established for many members of this class. This approach will closely follow similar results in the literature, including recent treatments by McCullough and Cole-Lewis-Wermer. Reproducing kernel Hilbert spaces where the analogue of the Nevanlinna-Pick theorem holds are particularly nice. In Chapter 5, the operator theory of these so-called complete Pick spaces will be developed, and used to tackle certain interpolation problems where additional constraints are imposed on the solution. A non-commutative view of interpola- tion will be presented, with the non-commutative analytic Toeplitz algebra of Popescu and Davidson-Pitts playing a prominent role. It is often useful to consider reproducing kernel Hilbert spaces which arise as natural products of other spaces. The Hardy space of the polydisk is the prime example of this. A general commutative and non-commutative view of such spaces will be presented in Chapter 6, using the left regular representation of higher-rank graphs, first introduced by Kribs-Power. A recent factorization theorem of Bercovici will be applied to these algebras, from which a Pick-type theorem may be deduced. The operator-valued Pick problem for these spaces will also be discussed. In Chapter 7, the various tools developed in this thesis will be applied to two related problems, known as the Douglas problem and the Toeplitz corona problem.
Lectures On Analytic Function Spaces And Their Applications
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Author : Javad Mashreghi
language : en
Publisher: Springer Nature
Release Date : 2023-11-14
Lectures On Analytic Function Spaces And Their Applications written by Javad Mashreghi and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-11-14 with Mathematics categories.
The focus program on Analytic Function Spaces and their Applications took place at Fields Institute from July 1st to December 31st, 2021. Hilbert spaces of analytic functions form one of the pillars of complex analysis. These spaces have a rich structure and for more than a century have been studied by many prominent mathematicians. They have essential applications in other fields of mathematics and engineering. The most important Hilbert space of analytic functions is the Hardy class H2. However, its close cousins—the Bergman space A2, the Dirichlet space D, the model subspaces Kt, and the de Branges-Rovnyak spaces H(b)—have also garnered attention in recent decades. Leading experts on function spaces gathered and discussed new achievements and future venues of research on analytic function spaces, their operators, and their applications in other domains. With over 250 hours of lectures by prominent mathematicians, the program spanned a wide variety of topics. More explicitly, there were courses and workshops on Interpolation and Sampling, Riesz Bases, Frames and Signal Processing, Bounded Mean Oscillation, de Branges-Rovnyak Spaces, Blaschke Products and Inner Functions, and Convergence of Scattering Data and Non-linear Fourier Transform, among others. At the end of each week, there was a high-profile colloquium talk on the current topic. The program also contained two advanced courses on Schramm Loewner Evolution and Lattice Models and Reproducing Kernel Hilbert Space of Analytic Functions. This volume features the courses given on Hardy Spaces, Dirichlet Spaces, Bergman Spaces, Model Spaces, Operators on Function Spaces, Truncated Toeplitz Operators, Semigroups of weighted composition operators on spaces of holomorphic functions, the Corona Problem, Non-commutative Function Theory, and Drury-Arveson Space. This volume is a valuable resource for researchers interested in analytic function spaces.
Nevanlinna Pick Interpolation On Sobolev Spaces With Boundary Conditions
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Author : Siamak Hashemi-Asasi
language : en
Publisher:
Release Date : 1991
Nevanlinna Pick Interpolation On Sobolev Spaces With Boundary Conditions written by Siamak Hashemi-Asasi and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1991 with categories.
The Dirichlet Space And Related Function Spaces
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Author : Nicola Arcozzi
language : en
Publisher: American Mathematical Soc.
Release Date : 2019-09-03
The Dirichlet Space And Related Function Spaces written by Nicola Arcozzi and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-09-03 with Dirichlet principle categories.
The study of the classical Dirichlet space is one of the central topics on the intersection of the theory of holomorphic functions and functional analysis. It was introduced about100 years ago and continues to be an area of active current research. The theory is related to such important themes as multipliers, reproducing kernels, and Besov spaces, among others. The authors present the theory of the Dirichlet space and related spaces starting with classical results and including some quite recent achievements like Dirichlet-type spaces of functions in several complex variables and the corona problem. The first part of this book is an introduction to the function theory and operator theory of the classical Dirichlet space, a space of holomorphic functions on the unit disk defined by a smoothness criterion. The Dirichlet space is also a Hilbert space with a reproducing kernel, and is the model for the dyadic Dirichlet space, a sequence space defined on the dyadic tree. These various viewpoints are used to study a range of topics including the Pick property, multipliers, Carleson measures, boundary values, zero sets, interpolating sequences, the local Dirichlet integral, shift invariant subspaces, and Hankel forms. Recurring themes include analogies, sometimes weak and sometimes strong, with the classical Hardy space; and the analogy with the dyadic Dirichlet space. The final chapters of the book focus on Besov spaces of holomorphic functions on the complex unit ball, a class of Banach spaces generalizing the Dirichlet space. Additional techniques are developed to work with the nonisotropic complex geometry, including a useful invariant definition of local oscillation and a sophisticated variation on the dyadic Dirichlet space. Descriptions are obtained of multipliers, Carleson measures, interpolating sequences, and multiplier interpolating sequences; estimates are obtained to prove corona theorems.
Function Spaces
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Author : Krzysztof Jarosz
language : en
Publisher: American Mathematical Soc.
Release Date : 2007
Function Spaces written by Krzysztof Jarosz and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Mathematics categories.
This book consists of contributions by the participants of the Fifth Conference on Function Spaces, held at Southern Illinois University in May of 2006. The papers cover a broad range of topics, including spaces and algebras of analytic functions of one and of many variables (and operators on such spaces), $L{p $-spaces, spaces of Banach-valued functions, isometries of function spaces, geometry of Banach spaces, and other related subjects. The goal of the conference was to bring together mathematicians interested in various problems related to function spaces and to facilitate the exchange of ideas between people working on similar problems. Hence, the majority of papers in this book are accessible to non-experts. Some articles contain expositions of known results and discuss open problems, others contain new results.