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Loewner's Theorem on Monotone Matrix Functions

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Cover of 'Loewner's Theorem on Monotone Matrix Functions'

Table of Contents

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    Book Overview
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    Chapter 1 Introduction: The Statement of Loewner’s Theorem
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    Chapter 2 Some Generalities
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    Chapter 3 The Herglotz Representation Theorems and the Easy Direction of Loewner’s Theorem
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    Chapter 4 Monotonicity of the Square Root
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    Chapter 5 Loewner Matrices
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    Chapter 6 Heinävaara’s Integral Formula and the Dobsch–Donoghue Theorem
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    Chapter 7 ℳ n + 1 ≠ ℳ n $${\mathcal M}_{n+1} \ne {\mathcal M}_n$$
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    Chapter 8 Heinävaara’s Second Proof of the Dobsch–Donoghue Theorem
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    Chapter 9 Convexity, I: The Theorem of Bendat–Kraus–Sherman–Uchiyama
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    Chapter 10 Convexity, II: Concavity and Monotonicity
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    Chapter 11 Convexity, III: Hansen–Jensen–Pedersen (HJP) Inequality
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    Chapter 12 Convexity, IV: Bhatia–Hiai–Sano (BHS) Theorem
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    Chapter 13 Convexity, V: Strongly Operator Convex Functions
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    Chapter 14 2 × 2 Matrices: The Donoghue and Hansen–Tomiyama Theorems
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    Chapter 15 Quadratic Interpolation: The Foiaş–Lions Theorem
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    Chapter 16 Pick Interpolation, I: The Basics
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    Chapter 17 Pick Interpolation, II: Hilbert Space Proof
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    Chapter 18 Pick Interpolation, III: Continued Fraction Proof
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    Chapter 19 Pick Interpolation, IV: Commutant Lifting Proof
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    Chapter 20 A Proof of Loewner’s Theorem as a Degenerate Limit of Pick’s Theorem
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    Chapter 21 Rational Approximation and Orthogonal Polynomials
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    Chapter 22 Divided Differences and Polynomial Approximation
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    Chapter 23 Divided Differences and Multipoint Rational Interpolation
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    Chapter 24 Pick Interpolation, V: Rational Interpolation Proof
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    Chapter 25 Loewner’s Theorem via Rational Interpolation: Loewner’s Proof
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    Chapter 26 The Moment Problem and the Bendat–Sherman Proof
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    Chapter 27 Hilbert Space Methods and the Korányi Proof
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    Chapter 28 The Krein–Milman Theorem and Hansen’s Variant of the Hansen–Pedersen Proof
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    Chapter 29 Positive Functions and Sparr’s Proof
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    Chapter 30 Ameur’s Proof Using Quadratic Interpolation
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    Chapter 31 One-Point Continued Fractions: The Wigner–von Neumann Proof
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    Chapter 32 Multipoint Continued Fractions: A New Proof
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    Chapter 33 Hardy Spaces and the Rosenblum–Rovnyak Proof
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    Chapter 34 Mellin Transforms: Boutet de Monvel’s Proof
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    Chapter 35 Loewner’s Theorem for General Open Sets
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    Chapter 36 Operator Means, I: Basics and Examples
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    Chapter 37 Operator Means, II: Kubo–Ando Theorem
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    Chapter 38 Lieb Concavity and Lieb–Ruskai Strong Subadditivity Theorems, I: Basics
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    Chapter 39 Lieb Concavity and Lieb–Ruskai Strong Subadditivity Theorems, II: Effros’ Proof
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    Chapter 40 Lieb Concavity and Lieb–Ruskai Strong Subadditivity Theorems, III: Ando’s Proof
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    Chapter 41 Lieb Concavity and Lieb–Ruskai Strong Subadditivity Theorems, IV: Aujla–Hansen–Uhlmann Proof
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    Chapter 42 Unitarily Invariant Norms and Rearrangement
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    Chapter 43 Unitarily Invariant Norm Inequalities
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Title
Loewner's Theorem on Monotone Matrix Functions
Published by
Springer International Publishing, October 2019
DOI 10.1007/978-3-030-22422-6
ISBNs
978-3-03-022421-9, 978-3-03-022422-6
Authors

Simon, Barry

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