Loewner's Theorem on Monotone Matrix Functions

• Κύριος συγγραφέας: Simon, Barry (Συγγραφέας, http://id.loc.gov/vocabulary/relators/aut)
• Συγγραφή απο Οργανισμό/Αρχή: SpringerLink (Online service)
• Μορφή: Ηλεκτρονική πηγή Ηλ. βιβλίο
• Γλώσσα: English
• Έκδοση: Cham : Springer International Publishing : Imprint: Springer, 2019.
• Έκδοση: 1st ed. 2019.
• Σειρά: Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics, 354
• Θέματα: Functions of real variables.; Matrix theory.; Algebra.; Operator theory.; Group theory.; Real Functions.; Linear and Multilinear Algebras, Matrix Theory.; Operator Theory.; Group Theory and Generalizations.

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 m...

Πίνακας περιεχομένων:

• 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.