Operator Theory, Operator Algebras and Their Interactions with Geometry and Topology: Ronald G. Douglas Memorial Volume

Preface
Contents
Following in the Footsteps of Ronald G. Douglas
References
Functional Models for Commuting Hilbert-Space Contractions
1 Introduction
2 Preliminaries
2.1 Notation
2.2 Domains with Motivation from Control: The Symmetrized Bidisk G and the Tetrablock E
2.3 Fundamental Operators
2.4 Models for Commutative Isometric Tuples
2.5 Canonical Commutative Unitary Tuple Associated with a Commutative Tuple of Contractions
3 Fundamental Operators for a Tuple of Commutative Contractions
4 Joint Halmos Dilation of Fundamental Operators
5 Non-commutative Isometric Lift of Tuples of Commutative Contractions
6 Pseudo-Commutative Contractive Lifts and Models for Tuples of Commutative Contractions
7 Characteristic Triple for a Tuple of Commutative Contractions
References
The Extended Aluthge Transform
1 Introduction
2 The Extended Aluthge Transform
3 Fixed Points of the Extended Aluthge Transform
4 The Case of T Idempotent
5 Some Useful Identities
6 An Application: The Spherical Aluthge Transform
7 Extended Aluthge Transforms of Complex Symmetric Operators
8 The Numerical Range and the Extended Aluthge Transform
8.1 Numerical Range for Extended Aluthge Transforms
References
Open Problems in Wavelet Theory
1 Introduction
2 One Dimensional Wavelets
2.1 Connectivity of Wavelets
2.2 Wavelets for H2(R)
2.3 Minimality of MSF Wavelets
2.4 Density of Riesz Wavelets
2.5 Intersection of Negative Dilates
2.6 Extension of Wavelet Frames
2.7 A Simple Question that Nobody has Bothered to Answer
3 Higher Dimensional Wavelets
3.1 Known Results
3.2 Characterization of Dilations
3.3 Calderón's Formula
3.4 Well-localized Wavelets
3.5 Meyer Wavelets for Integer Dilations
3.6 Schwartz Class Wavelets
4 Proof of Theorem 2.8
References
When Is Every Quasi-Multiplier a Multiplier?
1 Introduction
2 Preliminaries
3 Results and Concluding Remarks
References
Isomorphism in Wavelets II
1 Introduction
2 Definition of Isomorphisms
3 Proof of Theorem 2.5
4 Examples from Higher Dimensions to One Dimension
5 From Lower Dimensions to Higher Dimensions
References
Nevanlinna-Pick Families and Singular Rational Varieties
1 Introduction
2 The Conductor Ideal
3 Nevanlinna-Pick Families: The Scalar Case
4 Nevanlinna-Pick Families: The Matrix Case
5 Property A1(1) for Matrices
References
On Certain Commuting Isometries, Joint Invariant Subspaces and C*-Algebras
Notations
1 Introduction
2 Pure n-Isometries and Model Pure n-Isometries
3 Joint Invariant Subspaces
4 C*-Algebras Generated by Commuting Isometries
References
Spectral Analysis, Model Theory and Applications of Finite-Rank Perturbations
1 Introduction
1.1 Notation
2 Perturbation-Theoretic Background
3 Aspects of Self-Adjoint Rank-One Perturbations
3.1 Scales of Hilbert Spaces
3.2 Spectral Theory of Rank-One Perturbations
4 Aspects of Unitary Rank-One Perturbations and Model Theory
4.1 Aleksandrov–Clark Theory and Sz.-Nagy–Foiaş Model for Perturbations with Purely Singular Spectrum
4.2 de Branges–Rovnyak Model and Perturbations in the Extreme Case
4.3 General Perturbations and Nikolski–Vasyunin Model Theory
5 Self-Adjoint Finite-Rank Perturbations
5.1 Singular Finite-Rank Perturbations
5.2 Self-Adjoint Extensions
5.3 Some Applications of Singular Finite Rank Perturbations
6 Spectral Theory of Self-Adjoint Finite-Rank Perturbations
6.1 Absolutely Continuous Spectrum and Scattering Theory
6.2 Vector Mutually Singular Parts
6.3 Equivalence Classes and Spectral Multiplicity
7 Model Theory of Finite-Rank Unitary Perturbations
7.1 Setup and Model Spaces
7.2 Krein Spaces and Reproducing Kernel Hilbert Spaces in Applications
8 Spectral Theory of Finite-Rank Unitary Perturbations
8.1 Spectral Properties in Terms of the Characteristic Function
8.2 Singular Part in Terms of the Characteristic Function
Appendix: Brief Summaries of Other Closely Related Topics
Aleksandrov Spectral Averaging
Poltoratski's Theorem
Simon–Wolff Criterion
Functions of Several Variables
References
Invariance of the Essential Spectra of Operator Pencils
1 Introduction
2 Preliminaries on Linear Relations
3 Essential Spectra of the Operator Pencil λS-T and the Linear Relation TS-1
4 Essential Spectrum of Linear Relations Under Perturbations
5 Essential Spectrum of Operator Pencils Under Perturbations
References
Decomposition of the Tensor Product of Two Hilbert Modules
1 Introduction
1.1 Hilbert Module
1.2 The Jet Construction
2 A New Non-negative Definite Kernel
2.1 Boundedness of the Multiplication Operator on (H, K)
3 Realization of (H, K (α,β))
3.1 Description of the Hilbert Module S1
3.2 Description of the Quotient Module A1
4 Generalized Bergman Kernels
4.1 The Class FB2(Ω)
5 The Generalized Wallach Set
5.1 Generalized Wallach Set for the Bergman Kernel of the Euclidean Unit Ball in Cm
6 Quasi-Invariant Kernels
References
A Survey on Classification of C*-Algebras with the Ideal Property
1 Introduction
2 Background
3 Characterization of Simple, Real Rank Zero and Ideal Properties
4 Reduction Theorem
5 Extended Elliott Invariant and Stevens-Jiang Invariant
6 Classification of AH Algebras with the Ideal Property
References
A Survey on the Arveson-Douglas Conjecture
1 The Arveson-Douglas Conjecture
2 Backgrounds and Applications
2.1 Geometric Invariants for Row Contractions
2.2 A New Kind of Index Theorem
2.3 Connection with Holomorphic Extension Theorems
3 The Case of Low Dimensions or Co-dimensions
3.1 Principal Submodules
3.2 Case of Low Dimension
4 The Geometric Arveson-Douglas Conjecture
5 Decomposition of Modules and Varieties
5.1 Decomposition of Submodules
5.2 Decomposition of Quotient Modules
6 Quotient Modules over the Polydisc, Distinguished Varieties and Boundary Representations
References
The Pieri Rule for GLn Over Finite Fields
1 Introduction
1.1 Young Diagrams and Parabolic Subgroups
1.2 The Pieri Problem
2 Representations of Sn
2.1 The Young Modules
2.2 Properties of the Young Modules
2.3 The Irreducible Representations of Sn
2.4 The Grothendieck Group of Sn
3 Spherical Principal Series Representations of GLn
3.1 The Spherical Principal Series
3.2 The Grothendieck Group of the Spherical Principal Series
3.3 The Grothendieck Groups of Sn and of the Spherical Principal Series
4 The Pieri Rule
4.1 Skew-Diagrams and Horizontal Strips
4.2 The Pieri Rule for GLn(C )
4.3 Schur-Weyl Duality
4.4 The Pieri Rule for Sn
4.5 The Pieri Rule for GLn(Fq)
Appendix A Proofs
A.1 Proofs for Sect.2
Proof of Theorem 2.2.1
Proof of Corollary 2.3.1
A.2 Proofs for Sect.4
Proof of Theorem 4.4.1
References
Cauchy-Riemann Equations for Free Noncommutative Functions
1 Introduction
2 Real nc Functions Are nc Functions
3 Differentiability of nc Functions
4 Real and Imaginary Part of an nc Function
5 Cauchy-Riemann Equations: Sufficiency
References
Uniform Roe Algebras and Geometric RD Property
1 Introduction
2 Geometric Rapid Decay Spaces and Spectral Invariance
3 Examples of Geometrically Rapid Decay Spaces
4 Quasi-Isometry and Geometrically Rapid Decay Spaces
5 The Product Space of Geometric RD Spaces
6 The Fundamental Cyclic Cocycle
7 The Recent Developments
References
Integral Curvature and Similarity of Cowen-Douglas Operators
1 Introduction
2 Integral Curvature and Similarity of Operators in B1(Ω)
3 U+K Similarity of Operators in B1(Ω)
References
Singular Subgroups in A2-Groups and Their von Neumann Algebras
1 Introduction
2 Preliminaries and Main Theorem
2.1 Affine Buildings
2.2 A2-Groups
2.3 Singular Subgroups
2.4 Main Theorem and Its Proof
3 Acylindrically Hyperbolic Groups
4 The Case of No Loxodromic Elements in H
References
A K-Theoretic Selberg Trace Formula
1 Introduction
2 Statement of Selberg Trace Formula
3 Index Theoretic Trace Formula
3.1 Geometric Side
3.2 Spectral Side
4 K-Theoretic Selberg Trace Formula
4.1 Decomposition of Right Regular Representation of G
4.2 Decomposition of the Trivial Representation of
4.3 The Restriction Map
4.4 Selberg Trace Formula in K-Theory
References
Singular Hilbert Modules on Jordan–Kepler Varieties
1 Introduction
1.1 The Normalized Kernel
2 Invariants for Submodules
3 Jordan–Kepler Varieties
4 Hilbert Modules on Kepler Varieties
5 The Singular Set and Its Resolution
6 Singular Hilbert Submodules
7 Outlook and Concluding Remarks
References
A Survey of Ron Douglas's Contributions to the Index Theory of Toeplitz Operators
1 Introduction
2 Toeplitz Operators Associated to a Semigroup
3 Toeplitz Operators on the Real Line
4 Quarter-Plane Toeplitz Operators
5 Toeplitz Operators Associated to Elliptic Operators
6 Toeplitz Operators Associated to a Foliation
References
Differential Subalgebras and Norm-Controlled Inversion
1 Introduction
2 Differential Subalgebras
3 Generalized Differential Subalgebras
4 Brandenburg Trick and Norm-Controlled Inversion
References
Hermitian Metrics on the Resolvent Set and Extremal Arc Length
1 Introduction
2 Extremal Equation
3 The Unilateral Shift Operator
4 Energy Functional
References
Hybrid Normed Ideal Perturbations of n-Tuples of Operators II: Weak Wave Operators
1 Introduction
2 Existence of Generalized Wave Operators
3 Invariance of Lebesgue Absolutely Continuous Parts Under Perturbations
References
An Introduce to Curvature Inequalities for Operators in the Cowen–Douglas Class
1 Vector Bundle and Curvature Relevant to Operator
2 Curvature Inequities for Cowen–Douglas Operator
2.1 Curvature Inequalities of Order 1
2.2 Curvature Inequalities of Higher Order
2.3 Further Discussion
References
Reproducing Kernel of the Space Rt(K,μ)
1 Introduction
2 Proof of Main Theorem
References