Linked e-resources
Details
Table of Contents
Intro
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
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