Loewner's Theorem on Monotone Matrix FunctionsThis 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. |
Contents
| 1 | |
Part II Proofs of the Hard Direction | 160 |
Part III Applications and Extensions | 370 |
A Boutet de Monvels Note | 415 |
B Pictures | 422 |
C Symbol List | 427 |
| 432 | |
| 447 | |
| 453 | |
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Common terms and phrases
analytic continuation Blaschke product bounded Chapter Check for updates commutes compact continuous function convergence convex functions Corollary define divided differences Donoghue eigenvalues Equation equivalent extreme points finite follows formula function f given Grundlehren der mathematischen Herglotz representation Hilbert space Historical Remarks holds implies inequality integral Lemma Let f linear Loewner matrix Loewner's theorem mathematischen Wissenschaften 354 matrix convex functions matrix monotone matrix monotone functions measure Mellin transform Mn(a monotone functions Monotone Matrix Functions Neumann Notes and Historical obeys orthogonal Pick matrix Pick's theorem poles problem Proof Let proof of Loewner's Proof of Theorem Proposition prove Theorem rational function result Schur Schur functions self-adjoint operators shows Simon Springer Nature Switzerland strictly positive subset Suppose Switzerland AG 2019 Theorem on Monotone unitary zeros