Operator and Norm Inequalities and Related Topics

Preface
Part I: Matrix and Operator Inequalities
Part II: Orthogonality and Inequalities
Part III: Inequalities Related to Types of Operators
Part IV: Inequalities in Various Banach Spaces
Part V: Inequalities in Commutative and Noncommutative Probability Spaces
Contents
Part I Matrix and Operator Inequalities
Log-majorization Type Inequalities
1 Introduction
2 Matrix Majorization
3 Trace and Determinantal Inequalities
4 Golden-Thompson Inequality and Araki's Log-majorization
5 Ando-Hiai Inequality
6 BLP and Matharu-Aujla Inequalities
7 Inequalities for Operator Connections
8 Ando and Visick's Inequalities for the Hadamard Product
9 Indefinite Inequalities
References
Ando-Hiai Inequality: Extensions and Applications
1 Introduction
2 Extensions
3 Applications
4 Concluding Remarks
References
Relative Operator Entropy
1 Introduction
2 Operator Means and Solidarities
3 Relative Operator Entropy
4 CPR Geometry
5 Tsallis Relative Entropy
6 Concluding Remarks
References
Matrix Inequalities and Characterizations of Operator Monotone Functions
1 Introduction
2 Matrix Inequalities and Characterizations of Operator Functions
2.1 Heinz Mean, Heron Mean, and Operator Monotone Functions
2.1.1 Scalar Inequality for Heinz Mean and Heron Mean
2.1.2 Matrix Inequalities and Operator Monotone Functions
2.2 Symmetric and Self-adjoint Means via Integral Representations
2.2.1 Symmetric Means
2.2.2 Self-adjoint Means
2.2.3 Kubo-Ando Condition
2.2.4 General Symmetric Means
2.3 Matrix Power Means and Operator Monotone Functions
2.3.1 Kubo-Ando Matrix Power Means and Characterizations
2.3.2 The Inverse Problem for Non-Kubo-Ando Matrix Power Means
3 Powers-Størmer's Inequality and Characterizations of Operator Monotone Functions
References
Perspectives, Means and their Inequalities
1 Introduction
2 Perspectives for Invertible Operators
2.1 Homogeneity
2.2 Convexity
2.3 Monotonicity and Convergence
2.3.1 Monotonicity for One Direction
2.3.2 Monotonicity for Each Variable
3 An Extension of the Perspective Function
3.1 A Functional Calculus for Commuting Positive Operators
3.2 Pusz–Woronowicz Functional Calculus
3.2.1 The Commuting Pair (R,S)
3.2.2 Variational Expression
3.2.3 Pusz–Woronowicz Functional Calculus
3.2.4 Homogeneity, Upper Continuity and Convexity
3.2.5 Restricted Domain
4 Theory of Operator Means
4.1 Kubo-Ando's Axiomatization
4.2 Operator Means
4.2.1 Integral Representation
4.2.2 Geometric Mean
4.2.3 Mean of Projections
4.2.4 Transforms on OM+1
4.2.5 Weight and Symmetricity
4.2.6 Power Means
4.2.7 Stolarsky Means
4.2.8 Means of Szabó Type
5 Operator Inequalities
5.1 Positive Maps
5.2 Power Monotonicity
5.2.1 Ando–Hiai Type Inequalities
5.3 Furuta Inequality
5.4 Chaotic Order
5.5 Notes and Remarks
References
Cauchy–Schwarz Operator and Norm Inequalities for Inner Product Type Transformers in Norm Ideals of Compact Operators, with Applications
1 Introduction
2 Cauchy–Schwarz Operator and Norm Inequalities for i.p.t. Transformers
3 Aczél–Bellman Type Norm Inequalities for i.p.t. Transformers
4 Norm Inequalities for Transformers Generated by Analytic Functions with Non-negative Taylor Coefficients
5 Grüss–Landau Type Norm Inequalities for i.p.t. Transformers
6 Norm Inequalities for Holomorphic Functions on Simply Connected Domains in the Complex Plane
7 Mean Value Norm Inequalities for Operator Monotone Functions and Heinz Inequalities, with Applications
8 Laplace Transformers, Arithmetic-Geometric and Young Norm Inequalities
9 Applications to Refinements and Generalizations of Minkowski, Zhan and Heron Inequalities
10 Connections with Cauchy–Schwarz Inequalities for Hilbert Modules
References
Norm Estimations for the Moore-Penrose Inverse of the Weak Perturbation of Hilbert C*-Module Operators
1 Introduction
2 Some Basic Knowledge About the Moore-Penrose Inverse
3 Various Types of Perturbations
3.1 The Multiplicative Perturbation Case
3.2 The Stable Additive Perturbation and the Strong Perturbation
4 Representations for the Moore-Penrose Inverse
5 Norm Estimations for the Moore-Penrose Inverse
References
Part II Orthogonality and Inequalities
Birkhoff–James Orthogonality: Characterizations, Preservers, and Orthogonality Graphs
1 Introduction
1.1 Roberts Orthogonality
1.2 Isosceles Orthogonality
1.3 Birkhoff–James Orthogonality
1.4 A Plethora of Orthogonalities
2 Properties of B-J Orthogonality
2.1 Gateaux Derivatives and Complex Case Related to Real Orthogonality
2.2 Symmetry, Additivity and Uniqueness of B-J Orthogonality
2.3 Mutual B-J Orthogonality
3 Examples of B-J Orthogonality
3.1 B-J Orthogonality in B(H) and Beyond
3.2 B-J Orthogonality in Commutative C*-Algebras and Function Spaces
3.3 B-J Orthogonality in General C*-Algebras and Hilbert C*-Modules
3.4 Strong B-J Orthogonality on Hilbert C*-Modules
4 Applications of B-J Orthogonality
5 Preservers of B-J Orthogonality
6 Graph Induced by B-J Orthogonality
6.1 Property Recognition
6.2 Isomorphism Problem
References
Approximate Birkhoff-James Orthogonality in Normed Linear Spaces and Related Topics
1 Introduction and Preliminaries
1.1 Orthogonality in Inner Product Spaces
1.2 Orthogonalities in Normed Spaces
2 Approximate Birkhoff-James Orthogonality in Normed Linear Spaces
3 Applications and Generalizations—Review of Selected Results
3.1 Approximate Birkhoff-James Orthogonality in Operator Theory
3.2 Operators Approximately Preserving Orthogonality
3.3 Approximate Symmetry of the Birkhoff-James Orthogonality
3.4 Varia
References
Orthogonally Additive Operators on Vector Lattices
1 Introduction
2 Definition and Main Examples of OAOs
3 The Lateral Order and Related Notions
3.1 Basic Properties
3.2 Lateral Ideal and Lateral Bands
3.3 The Intersection Property
4 Extension of Orthogonally Additive Maps
5 The Order Structure on the Vector Lattice of OAOs
5.1 Order Calculus and Riesz-Kantorovich Types Formulas
5.2 The Boolean Algebra of Fragments of a Positive OAO
6 Compact Orthogonally Additive Operators
6.1 The Projection Band of C-Compact Orthogonally Additive Operators
6.2 Domination Problem for AM-Compact Abstract Uryson Operator
7 Partial Order Continuities of Orthogonally Additive Operators
8 Narrow OAOs and Representation of Regular Operators
8.1 Narrow Operators
8.2 Representation of Regular Operators
9 Banach Lattices of OAOs
9.1 Absolute and Uniform Norms of an Abstract Uryson Operator
9.2 Consistent Sets and Levels in a Vector Lattice
9.3 Linear Sections of Orthogonally Additive Operators
10 Open Problems
10.1 An Analytic Representation of OAOs
10.2 Disjointness Preserving OAOs
10.3 Compact OAOs
10.4 Order Projections
10.5 Partial Order Continuities of Orthogonally Additive Operators
References
Part III Inequalities Related to Types of Operators
Normal Operators and their Generalizations
1 Introduction
2 Notations and Preliminary Results on Spectral Theory
3 Some Notions of Local Spectral Theory
4 Normal Type Operators
5 Totally Hereditarily Normaloid Operators
6 Weyl Type Theorems for Analytically Quasi THN Operators
References
On Wold Type Decomposition for Closed Range Operators
1 Introduction
1.1 Beurling Theorems for Shift Operators
1.1.1 The Hardy Space
1.1.2 The Bergman Space
1.1.3 The Direchlet Space
1.1.4 More on Beurling's Theorem for Hilbert Spaces of Analytic Functions
1.2 Beurling's Type Theorem for Left Invertible Operators Close to Isometries
2 Regular Operators
2.1 Moore-Penrose Generalized Inverse
2.2 Some Basic Properties of Regular Operators
2.3 Restrictions of Regular Operators
3 Wold Type Decomposition for Regular Operators
3.1 The Main Results
3.2 Applications to Weighted Shifts
4 Wold-Type Decomposition for Bi-Regular Operators
4.1 First Properties
4.2 The Cauchy Dual of a Closed Range Operator
4.3 Extended Wold-Type Decomposition
5 Some Open Questions
References
(Asymmetric) Dual Truncated Toeplitz Operators
1 Motivation and Basic Notations
2 Restrictions of Multiplication Operators and Its Basic Properties
3 Basic Properties of ADTTO
4 Intertwining Property for ADTTO
5 Other Relations with ADTTO
6 A Characterization of ADTTO
7 A Brown–Halmos Type Theorem for DTTO
References
Boundedness of Toeplitz Operators in Bergman-Type Spaces
1 Introduction: The Spaces and Operators
2 Toeplitz Operators with Oscillating Symbols
3 Toeplitz Operators in Hv∞-Spaces: Introduction
4 Toeplitz Operators with Harmonic Symbols in Hv∞(D)-Spaces
5 General Result on Multipliers and Toeplitz Operators in Hv∞(D) with Radial Symbols
6 Supplementary Results on Toeplitz Operators with Radial Symbols
References
Part IV Inequalities in Various Banach Spaces
Disjointness Preservers and Banach-Stone Theorems
1 Introduction
2 Three Classical Theorems
3 Isomorphism of Disjointness Structure
3.1 -Isomorphisms
3.2 Homeomorphism Associated with a -Isomorphism
3.3 Representation
4 Applications
4.1 Order Isomorphism
4.2 Realcompact Spaces
4.3 Ring Isomorphism and Multiplicative Isomorphism
4.4 Isometry
4.5 Nonvanishing Preservers
References
The Bishop–Phelps–Bollobás Theorem: An Overview
1 Motivation and Historical Background
2 The Bishop–Phelps–Bollobás Property
2.1 For Operators
2.2 For Some Classes of Operators
2.3 For Multilinear Mappings
2.4 For Homogeneous Polynomials
2.5 For Holomorphic Functions
2.6 For Lipschitz Mappings
2.7 For Numerical Radius
3 Sharpness: The Bishop–Phelps–Bollobás Moduli
4 The Point and Operator Properties
5 The Local Properties
5.1 Local Properties for Operators
5.2 Local Properties for Bilinear Mappings
6 Open Questions
7 Further Research and New (or Recent) Possible Lines
8 Tables for Classical Banach Spaces: A Summary
References
A New Proof of the Power Weighted Birman–Hardy–RellichInequalities
1 Introduction
2 Power-Weighted Birman–Hardy–Rellich Inequalities
3 Some Generalizations
Appendix A: Background for the Vector-Valued Case
References
An Excursion to Multiplications and Convolutionson Modulation Spaces
1 Introduction
2 Preliminaries
2.1 The Short-Time Fourier Transform
2.2 STFT Projections and a Suitable Twisted Convolution
2.3 Gelfand-Shilov Spaces
2.4 Weight Functions
2.5 Mixed Norm Spaces of Lebesgue Type
2.6 Convolutions and Multiplications for Discrete Lebesgue Spaces
3 Modulation Spaces, Multiplications and Convolutions
3.1 Modulation Spaces
3.2 Multiplications and Convolutions in Modulation Spaces
4 Gabor Products and Modulation Spaces
Appendix A: Some Properties of Wiener Amalgam Spaces
References
The Hardy-Littlewood Inequalities in Sequence Spaces
1 Introduction
2 Preliminary Results
3 Theorem 1.1 for m-Linear Forms
4 Theorem 1.2 for m-Linear Forms
5 Theorem 1.3 for Non-negative m-Linear Forms
6 Theorem 1.4 for m-Linear Forms
7 The Critical and Supercritical Cases
8 On the Constants and Some Final Remarks
9 The Gale-Berlekamp Switching Game
References
Symmetries of C-algebras and Jordan Morphisms
1 Introduction
2 Wigner Theorem
2.1 Probability Version
2.2 Logical Version
3 Dye Theorem
3.1 Dye Theorem for von Neumann Algebras
3.2 Dye Theorem for AW*-algebras
4 Structure of Abelian Subalgebras
4.1 Abelian C-subalgebras
4.2 Abelian von Neumann Subalgebras
4.3 Abelian Subalgebras as Invariants
5 Choquet Order Structure
References
Part V Inequalities in Commutative and Noncommutative Probability Spaces
Mixed Norm Martingale Hardy Spaces and Applications in Fourier Analysis
1 Introduction
2 Mixed Lebesgue Spaces
3 Doob's Inequality
4 Mixed Martingale Hardy Spaces
5 Atomic Decomposition
6 Martingale Inequalities
7 Dual Spaces of Mixed Hardy Spaces
8 One-dimensional Walsh-Fourier Series
9 Higher Dimensional Walsh-Fourier Series
References
The First Eigenvalue for Nonlocal Operators
1 Nonlocal Diffusion Problems
2 The First Eigenvalue with Dirichlet or Neumann Boundary Conditions
2.1 Dirichlet Boundary Conditions
2.2 Neumann Boundary Conditions
2.3 Optimal Constants in Lq
2.3.1 The Dirichlet Case
2.3.2 The Neumann Case
References
Comparing Banach Spaces for Systems of Free Random Variables Followed by the Semicircular Law
1 Introduction
1.1 Background
1.2 Motivation
1.3 Overview
2 Preliminaries
3 Free Distributions of Multi Semicircular Elements
4 A C*-Probability Space Generated by |N|-Many Semicircular Elements
4.1 Free-Isomorphic Relations
4.2 A C*-Probability Space Xφ
5 Free-Distributional Data on Xφ
6 Certain Free-Isomorphisms on Xφ
6.1 Shifts on Z
6.2 Integer Shifts on Xφ
6.3 Free-Isomorphic Relations on Xφ
7 Free Random Variables followed by the Semicircular Law
7.1 Group C*-Algebra of λ
7.2 The Tensor Product C*-Algebra X
7.3 Free-Distributional Information on Xτ
7.4 A Structure Theorem of Xτ
8 Certain Banach-Space Operators Acting on Xτ
8.1 Banach-Space Operators Ts,ltB(Xτ)
8.2 Multiplication Operators Muk0,j0 of B(Xτ)
References