Loewners Theorem on Monotone Matrix Functions

This book provides an in depth discussion of Loewner’s theorem on the characterization of matrix monotone functions. The author refers to the book as a ‘love poem,’ one that highlights a unique mix of algebra and analysis and touches on numerous methods and results. The book details many different topics from analysis, operator theory and algebra, such as divided differences, convexity, positive definiteness, integral representations of function classes, Pick interpolation, rational approximation, orthogonal polynomials, continued fractions, and more. Most applications of Loewner’s theorem involve the easy half of the theorem. A great number of interesting techniques in analysis are the bases for a proof of the hard half. Centered on one theorem, eleven proofs are discussed, both for the study of their own approach to the proof and as a starting point for discussing a variety of tools in analysis. Historical background and inclusion of pictures of some of the main figures who have developed the subject, adds another depth of perspective.The presentation is suitable for detailed study, for quick review or reference to the various methods that are presented. The book is also suitable for independent study. The volume will be of interest to research mathematicians, physicists, and graduate students working in matrix theory and approximation, as well as to analysts and mathematical physicists.

Preface.- Part I. Tools.- 1. Introduction: The Statement of Loewners Theorem.- 2. Some Generalities.- 3. The Herglotz Representation Theorems and the Easy Direction of Loewners Theorem.- 4. Monotonicity of the Square Root.- 5. Loewner Matrices.- 6. Heinävaaras Integral Formula and the Dobsch–Donoghue Theorem.- 7. Mn+1 ¹ Mn.- 8. Heinävaaras 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 Foiaş–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 Loewners Theorem as a Degenerate Limit of Picks 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. Loewners Theorem Via Rational Interpolation: Loewners Proof .- 26. The Moment Problem and the Bendat–Sherman Proof.- 27. Hilbert Space Methods and the Korányi Proof.- 28. The Krein–Milman Theorem and Hansens Variant of the Hansen–Pedersen Proof .- 29. Positive Functions and Sparrs Proof.- 30. Ameurs 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 Monvels Proof.- 35. Loewners 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: Andos 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 Monvels Note.- Appendix B. Pictures.- Appendix C. Symbol List.- Bibliography.- Author Index.- Subject Index.