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Contents

Preface

Chapter 1.  Positive Matrices

1.1   Characterizations

1.2   Some Basic Theorems

1.3   Block Matrices

1.4   Norm of the Schur Product

1.5   Monotonicity and Convexity

1.6   Supplementary Results and Exercises

1.7   Notes and References

Chapter 2.  Positive Linear Maps

2.1   Representations

2.2   Positive Maps

2.3   Some Basic Properties of Positive Maps

2.4   Some Applications

2.5   Three Questions

2.6   Positive Maps on Operator Systems

2.7   Supplementary Results and Exercises

2.8   Notes and References

Chapter 3.  Completely Positive Maps

3.1   Some Basic Theorems

3.2   Exercises

3.3   Schwarz Inequalities

3.4   Positive Completions and Schur Products

3.5   The Numerical Radius

3.6   Supplementary Results and Exercises

3.7   Notes and References

Chapter 4.  Matrix Means

4.1   The Harmonic Mean and the Geometric Mean

4.2   Some Monotonicity and Convexity Theorems

4.3   Some Inequalities for Quantum Entropy

4.4   Furuta’s Inequality

4.5   Supplementary Results and Exercises

4.6   Notes and References

Chapter 5.  Positive Definite Functions

5.1   Basic Properties

5.2   Examples

5.3   Loewner Matrices

5.4   Norm Inequalities for Means

5.5   Theorems of Herglotz and Bochner

5.6   Supplementary Results and Exercises

5.7   Notes and References

Chapter 6.  Geometry of Positive Matrices

6.1   The Riemannian Metric

6.2   The Metric Space

6.3   Center of Mass and Geometric Mean

6.4   Related Inequalities

6.5   Supplementary Results and Exercises

6.6   Notes and References

Bibliography

Index

Notation