Loewner's Theorem on Monotone Matrix Functions
Description
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.
Reviews / Votes
"This book will be a valuable reference for anyone interested in any aspect of Loewner's theorem. The variety of techniques used in the eleven proofs also makes the text a good introduction to many standard methods in functional analysis and function theory." (Linda J. Patton, Mathematical Reviews, October, 2020)"Doubtless, this 43-chapter book is very well written in a reader-friendly style. Chapters include some historical remarks and helpful comments. The reviewer would like to recommend the book strongly to postgraduate students and mathematicians interested in operator inequalities." (Mohammad Sal Moslehian, zbMATH 1428.26002, 2020)
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Content
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.