Loewner's Theorem on Monotone Matrix Functions

  • Barry Simon

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 354)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Part I

  3. Part II

    1. Front Matter
      Pages 161-164
    2. Barry Simon
      Pages 165-173
    3. Barry Simon
      Pages 301-312
    4. Barry Simon
      Pages 359-369
  4. Part III

  5. Back Matter
    Pages 415-459

About this book


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.


matrix convex approximation theory Pick interpolation Cauchy interpolation Loewner's theorem monotone matrix function Herglotz representation Simon Grundlehren Lieb concavity Donoghue theorem

Authors and affiliations

  • Barry Simon
    • 1
  1. 1.Division of Physics, Math, and AstronomyCaltechPasadenaUSA

Bibliographic information