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 Foias-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.