دانلود کتاب Loewner's theorem on monotone matrix functions

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Preface......Page 6
Contents......Page 10
Part I Tools......Page 13
1 Introduction: The Statement of Loewner\'s Theorem......Page 15
2 Some Generalities......Page 22
3 The Herglotz Representation Theorems and the Easy Direction of Loewner\'s Theorem......Page 28
4 Monotonicity of the Square Root......Page 43
5 Loewner Matrices......Page 52
6 Heinävaara\'s Integral Formula and the Dobsch–Donoghue Theorem......Page 81
7 Mn+1 =Mn......Page 92
8 Heinävaara\'s Second Proof of the Dobsch–Donoghue Theorem......Page 96
9 Convexity, I: The Theorem of Bendat–Kraus–Sherman–Uchiyama......Page 101
10 Convexity, II: Concavity and Monotonicity......Page 112
11 Convexity, III: Hansen–Jensen–Pedersen (HJP) Inequality......Page 122
12 Convexity, IV: Bhatia–Hiai–Sano (BHS) Theorem......Page 128
13 Convexity, V: Strongly Operator Convex Functions......Page 140
14 2 2 Matrices: The Donoghue and Hansen–Tomiyama Theorems......Page 150
15 Quadratic Interpolation: The Foiaş–Lions Theorem......Page 157
Part II Proofs of the Hard Direction......Page 164
16 Pick Interpolation, I: The Basics......Page 168
17 Pick Interpolation, II: Hilbert Space Proof......Page 177
18 Pick Interpolation, III: Continued Fraction Proof......Page 184
19 Pick Interpolation, IV: Commutant Lifting Proof......Page 190
20 A Proof of Loewner\'s Theorem as a Degenerate Limit of Pick\'s Theorem......Page 216
21 Rational Approximation and Orthogonal Polynomials......Page 222
22 Divided Differences and Polynomial Approximation......Page 235
23 Divided Differences and Multipoint Rational Interpolation......Page 246
24 Pick Interpolation, V: Rational Interpolation Proof......Page 261
25 Loewner\'s Theorem via Rational Interpolation: Loewner\'s Proof......Page 265
26 The Moment Problem and the Bendat–Sherman Proof......Page 271
27 Hilbert Space Methods and the Korányi Proof......Page 281
28 The Krein–Milman Theorem and Hansen\'s Variant of the Hansen–Pedersen Proof......Page 286
29 Positive Functions and Sparr\'s Proof......Page 297
30 Ameur\'s Proof Using Quadratic Interpolation......Page 309
31 One-Point Continued Fractions: The Wigner–von Neumann Proof......Page 317
32 Multipoint Continued Fractions: A New Proof......Page 329
33 Hardy Spaces and the Rosenblum–Rovnyak Proof......Page 334
34 Mellin Transforms: Boutet de Monvel\'s Proof......Page 340
35 Loewner\'s Theorem for General Open Sets......Page 353
Part III Applications and Extensions......Page 364
36 Operator Means, I: Basics and Examples......Page 365
37 Operator Means, II: Kubo–Ando Theorem......Page 371
38 Lieb Concavity and Lieb–Ruskai Strong Subadditivity Theorems, I: Basics......Page 377
39 Lieb Concavity and Lieb–Ruskai Strong Subadditivity Theorems, II: Effros\' Proof......Page 384
40 Lieb Concavity and Lieb–Ruskai Strong Subadditivity Theorems, III: Ando\'s Proof......Page 386
41 Lieb Concavity and Lieb–Ruskai Strong Subadditivity Theorems, IV: Aujla–Hansen–Uhlmann Proof......Page 388
42 Unitarily Invariant Norms and Rearrangement......Page 390
43 Unitarily Invariant Norm Inequalities......Page 399
Fonctions croissantes hilbertiennes......Page 405
Interpolation Functions......Page 409
B Pictures......Page 412
C Symbol List......Page 415
Bibliography......Page 420
Name Index......Page 434
Subject Index......Page 439