Loewner's theorem on monotone matrix functions
 Responsibility
 Barry Simon
 Publication
 Cham, Switzerland : Springer, [2019]
 Copyright notice
 ©2019
 Physical description
 xi, 459 pages : illustrations, portraits ; 25 cm
 Series
 Grundlehren der mathematischen Wissenschaften 354.
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QA331.5 .S56 2019  Unknown 
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Description
Creators/Contributors
 Author/Creator
 Simon, Barry, 1946 author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 433446) and indexes
 Contents

 Tools
 Introduction : The Statement of Loewnerʼs Theorem
 Some Generalities
 The Herglotz Representation Theorems and the Easy Direction of Loewnerʼs Theorem
 Monotonicity of the Square Root
 Loewner Matrices
 Heinävaaraʼs Integral Formula and the DobschDonoghue Theorem
 Mn+1 ... Mn
 Heinävaaraʼs Second Proof of the DobschDonoghue Theorem
 Convexity, I : The Theorem of BendatKrausShermanUchiyama
 Convexity, II : Concavity and Monotonicity
 Convexity, III : HansenJensenPedersen (HJP) Inequality
 Convexity, IV : BhatiaHiaiSano (BHS) Theorem
 Convexity, V : Strongly Operator Convex Functions
 2 x 2 Matrices : The Donoghue and HansenTomiyama Theorems
 Quadratic Interpolation : The FoiașLions Theorem
 Proofs of the Hard Direction
 Pick Interpolation, I : The Basics
 Pick Interpolation, II : Hilbert Space Proof
 Pick Interpolation, III : Continued Fraction Proof
 Pick Interpolation, IV : Commutant Lifting Proof
 A Proof of Loewnerʼs Theorem as a Degenerate Limit of Pickʼs Theorem
 Rational Approximation and Orthogonal Polynomials
 Divided Differences and Polynomial Approximation
 Divided Differences and Multipoint Rational Interpolation
 Pick Interpolation, V : Rational Interpolation Proof
 Loewnerʼs Theorem via Rational Interpolation : Loewnerʼs Proof
 The Moment Problem and the BendatSherman Proof
 Hilbert Space Methods and the Korányi Proof
 The KreinMilman Theorem and Hansenʼs Variant of the HansenPedersen Proof
 Positive Functions and Sparrʼs Proof
 Ameurʼs Proof Using Quadratic Interpolation
 OnePoint Continued Fractions : The Wignervon Neumann Proof
 Multipoint Continued Fractions : A New Proof
 Hardy Spaces and the RosenblumRovnyak Proof
 Mellin Transforms : Boutet de Monvelʼs Proof
 Loewnerʼs Theorem for General Open Sets
 Applications and Extensions
 Operator Means, I : Basics and Examples
 Operator Means, II : KuboAndo Theorem
 Lieb Concavity and LiebRuskai Strong Subadditivity : Theorems, I : Basics
 Lieb Concavity and LiebRuskai Strong Subadditivity : Theorems, II : Effros' Proof
 Lieb Concavity and LiebRuskai Strong Subadditivity : Theorems, III : Andoʼs Proof
 Lieb Concavity and LiebRuskai Strong Subadditivity : Theorems, IV : AujlaHansenUhlmann Proof
 Unitarily Invariant Norms and Rearrangement
 Unitarily Invariant Norm Inequalities
 Boutet de Monvelʼs Note
 Pictures
 Symbol List
 Bibliography
 Name Index
 Subject Index
 Summary

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.
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Bibliographic information
 Publication date
 2019
 Copyright date
 2019
 Series
 Grundlehren der mathematischen Wissenschaften = A series of comprehensive studies in mathematics, 00727830 ; volume 354
 ISBN
 9783030224219 hardbound
 9783030224226 ebook
 303022421X (hardcover)