Loewner's Theorem on Monotone Matrix Functions (Paperback)

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. Heinavaara's Integral Formula and the Dobsch?Donoghue Theorem.- 7. Mn+1 ¹ Mn.- 8. Heinavaara'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 Koranyi 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.