Loewner's Theorem on Monotone Matrix Functions

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  anda 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.

• 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 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 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.

“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)