Loewner's 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.

Auteur
Barry Simon is the IBM Professor of Mathematics and Theoretical Physics, Emeritus, at Caltech, known for his contributions in spectral theory, functional analysis, and nonrelativistic quantum mechanics, and including the connections to atomic and molecular physics. He has authored more than 400 publications on mathematics and physics. Simon's work has focused on broad areas of mathematical physics and analysis covering: quantum field theory, statistical mechanics, Brownian motion, random matrix theory, general nonrelativistic quantum mechanics, nonrelativistic quantum mechanics in electric and magnetic fields, the semi-classical limit, the singular continuous spectrum, random and ergodic Schrödinger operators, orthogonal polynomials, and non-selfadjoint spectral theory. Simon is a recently elected member (2019) of the National Academy of Science and a member of the American Academy of Arts and Sciences. Simon is a recipient of the Henri Poincaré Prize (2012), the Bolyai Prize of the Hungarian Academy of Sciences (2015), the Steele Prize for Lifetime achievements (2016), and the Dannie Heineman Prize for Mathematical Physics from the American Physical Society (2018). He is also a fellow of the American Mathematical Society and the American Physical Society.

Contenu

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 DobschDonoghue Theorem.- 7. Mn+1 ¹ Mn.- 8. Heinävaara's Second Proof of the DobschDonoghue Theorem.- 9. Convexity, I: The Theorem of BendatKrausShermanUchiyama.- 10. Convexity, II: Concavity and Monotonicity.- 11. Convexity, III: HansenJensenPedersen (HJP) Inequality.- 12. Convexity, IV: BhatiaHiaiSano (BHS) Theorem.- 13. Convexity, V: Strongly Operator Convex Functions.- 14. 2 x 2 Matrices: The Donoghue and HansenTomiyama Theorems.- 15. Quadratic Interpolation: The FoiaLions 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 BendatSherman Proof.- 27. Hilbert Space Methods and the Korányi Proof.- 28. The KreinMilman Theorem and Hansen's Variant of the HansenPedersen Proof .- 29. Positive Functions and Sparr's Proof.- 30. Ameur's Proof using Quadratic Interpolation.- 31. One-Point Continued Fractions: The Wignervon Neumann Proof.- 32. Multipoint Continued Fractions: A New Proof .- 33. Hardy Spaces and the RosenblumRovnyak 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: KuboAndo Theorem.- 38. Lieb Concavity and LiebRuskai Strong Subadditivity Theorems, I: Basics.- 39. Lieb Concavity and LiebRuskai Strong Subadditivity Theorems, II: Effros' Proof.- 40. Lieb Concavity and LiebRuskai Strong Subadditivity Theorems, III: Ando's Proof .- 41. Lieb Concavity and LiebRuskai Strong Subadditivity Theorems, IV: AujlaHansenUhlmann 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.