Loewner's Theorem on Monotone Matrix Functions

Preface. - Part I. Tools. - 1. Introduction: The Statement of Loewner' s Theorem. - 2. Some Generalities. - 3. The Herglotz Representation Theorems and the Easy Direction of Loewner' s Theorem. - 4. Monotonicity of the Square Root. - 5. Loewner Matrices. - 6. Heinävaara' s Integral Formula and the Dobsch Donoghue Theorem. - 7. Mn+1 ¹ Mn. - 8. Heinävaara' s Second Proof of the Dobsch Donoghue Theorem. - 9. Convexity, I: The Theorem of Bendat Kraus Sherman Uchiyama. - 10. Convexity, II: Concavity and Monotonicity. - 11. Convexity, III: Hansen Jensen Pedersen (HJP) Inequality. - 12. Convexity, IV: Bhatia Hiai Sano (BHS) Theorem. - 13. Convexity, V: Strongly Operator Convex Functions. - 14. 2 x 2 Matrices: The Donoghue and Hansen Tomiyama Theorems. - 15. Quadratic Interpolation: The Foia Lions Theorem. - Part II. Proofs of the Hard Direction. - 16. Pick Interpolation, I: The Basics. - 17. Pick Interpolation, II: Hilbert Space Proof. - 18. Pick Interpolation, III: Continued Fraction Proof. - 19. Pick Interpolation, IV: Commutant Lifting Proof. - 20. A Proof of Loewner' s Theorem as a Degenerate Limit of Pick' s Theorem. - 21. Rational Approximation and Orthogonal Polynomials. - 22. Divided Differences and Polynomial Approximation. - 23. Divided Differences and Multipoint Rational Interpolation. - 24. Pick Interpolation, V: Rational Interpolation Proof . - 25. Loewner' s Theorem Via Rational Interpolation: Loewner' s Proof . - 26. The Moment Problem and the Bendat Sherman Proof. - 27. Hilbert Space Methods and the Korányi Proof. - 28. The Krein Milman Theorem and Hansen' s Variant of the Hansen Pedersen Proof . - 29. Positive Functions and Sparr' s Proof. - 30. Ameur' s Proof using Quadratic Interpolation. - 31. One-Point Continued Fractions: The Wigner von Neumann Proof. - 32. Multipoint Continued Fractions: A New Proof . - 33. Hardy Spaces and the Rosenblum Rovnyak Proof. - 34. Mellin Transforms: Boutet de Monvel' s Proof. - 35. Loewner' s Theorem for General Open Sets. - Part III. Applications and Extensions. - 36. Operator Means, I: Basics and Examples. - 37. Operator Means, II: Kubo Ando Theorem. - 38. Lieb Concavity and Lieb Ruskai Strong Subadditivity Theorems, I: Basics. - 39. Lieb Concavity and Lieb Ruskai Strong Subadditivity Theorems, II: Effros' Proof. - 40. Lieb Concavity and Lieb Ruskai Strong Subadditivity Theorems, III: Ando' s Proof . - 41. Lieb Concavity and Lieb Ruskai Strong Subadditivity Theorems, IV: Aujla Hansen Uhlmann Proof. - 42. Unitarily Invariant Norms and Rearrangement . - 43. Unitarily Invariant Norm Inequalities. - Part IV. End Matter. - Appendix A. Boutet de Monvel' s Note. - Appendix B. Pictures. - Appendix C. Symbol List. - Bibliography. - Author Index. - Subject Index.