J-Contractive Matrix Valued Functions and Related Topics (Encyclopedia of Mathematics and its Applications)

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Dedication......Page 6
Contents\0......Page 8
Preface\0......Page 12
1 Introduction\0......Page 14
2.1 The classes P_const(J) and U_const (J)\0......Page 26
2.2 The Potapov-Ginzburg transform\0......Page 27
2.3 Linear fractional transformations\0......Page 37
2.4 Matrix balls\0......Page 43
2.5 Redheffer transforms\0......Page 47
2.6 Indefinite metrics in the space C^m\0......Page 50
2.7 Minus matrices\0......Page 58
2.8 Strictly minus matrices\0......Page 61
2.9 Linear fractional transformations in S^pxq_const\0......Page 71
2.10 Linear fractional transformations in C^pxq_const\0......Page 76
2.11 Transformations from S^pxq_const  into C^pxq_const\0......Page 79
2.12 Affine generalizations\0......Page 81
2.13 The J modulus\0......Page 85
2.14 Infinite products of matrices\0......Page 90
2.15 Some useful inequalities\0......Page 93
2.16 Bibliographical notes\0......Page 94
3.1 Basic classes of functions\0......Page 96
3.2 The Riesz-Herglotz-Nevanlinna representation\0......Page 100
3.3 Some subclasses of the Carath￠￥eodory class C^pxq\0......Page 104
3.4 Inner-outer factorization in the class H_\\infty\0......Page 106
3.5 Factorization in the classes N_+ and N\0......Page 111
3.6 The rank of meromorphic mvf?ˉs\0......Page 116
3.7 Inner and outer mvf?ˉs in H^pxq\0......Page 117
3.8 Fourier transforms and Paley-Wiener theorems\0......Page 118
3.9 The Beurling-Lax theorem\0......Page 120
3.10 Inner-outer factorization in H^pxq\0......Page 127
3.11 The Smirnov maximum principle\0......Page 131
3.12 Characterization via multiplication operators\0......Page 135
3.13 Factorization in N^pxq and denominators\0......Page 139
3.14 Some properties of outer mvf?ˉs\0......Page 146
3.15 Scalar denominators\0......Page 149
3.16 Factorization of positive semidefinite mvf?ˉs\0......Page 151
3.17 Blaschke-Potapov products in S^pxp\0......Page 155
3.18 Entire matrix functions of class N^pxq\0......Page 160
3.19 Entire inner mvf?ˉs\0......Page 164
3.20 Minimal denominators of mvf?ˉs f ?? E N^pxq\0......Page 168
3.21 The class II^pxq\0......Page 170
3.22 Mean types for mvf?ˉs in N^pxq and ￥D^pxq\0......Page 176
3.23 Bibliographical notes\0......Page 179
4.1 The classes P(J) and U(J)\0......Page 182
4.2 Blaschke-Potapov products\0......Page 187
4.3 Entire J-inner mvf?ˉs\0......Page 194
4.4 Multiplicative representations in the class P.(J)\0......Page 198
4.5 The classes P(jpq) and U(jpq )\0......Page 200
4.6 Associated pairs of the first kind\0......Page 203
4.7 The classes P(Jp ), P(jp, Jp ), U(Jp) and U(jp, Jp )\0......Page 208
4.8 Associated pairs of the second kind\0......Page 211
4.9 Singular and regular J-inner mvf?ˉs\0......Page 214
4.10 Domains of holomorphy of J inner mvf?ˉs\0......Page 220
4.11 Growth constraints on J-inner mvf?ˉs\0......Page 221
4.12 Monotonic sequences in P.(J)\0......Page 222
4.13 Linear fractional transformations\0......Page 224
4.14 Connections between the two kinds of associated pairs\0......Page 230
4.15 Redheffer transforms\0......Page 231
4.16 Strongly regular J-inner mvf?ˉs\0......Page 232
4.17 Minus matrix valued functions\0......Page 234
4.18 More on linear fractional transformations\0......Page 243
4.19 Bibliographical notes\0......Page 250
5.1 Preliminaries\0......Page 253
5.2 Examples\0......Page 260
5.3 A characterization of the spaces H(U)\0......Page 267
5.4 A finite dimensional example and extensions thereof\0......Page 274
5.5 Extension of H(U) into C for U ?? P.(J)\0......Page 279
5.6 The space H(b) for pxp inner mvf?ˉs b(￥?)\0......Page 283
5.7 The space H.(b) for pxp inner mvf?ˉs b(￥?)\0......Page 288
5.8 The space H(U) for mxp J-inner mvf?ˉs U(￥?)\0......Page 289
5.9 The space H(W) for W ?? U(jpq )\0......Page 302
5.10 The de Branges space B(E)\0......Page 306
5.11 Regular de Branges matrices E and spaces B(E)\0......Page 311
5.12 Connections between A and E\0......Page 313
5.13 A factorization and parametrization of mvf?ˉs A ?? U(Jp ).\0......Page 315
5.14 A description of H(W) ?? Lm2\0......Page 321
5.15 Characterizations of the classes US (J), UrR (J) and UrsR(J)\0......Page 326
5.17 Formulas for W(￥?) when W ?? UrsR(jpq )\0......Page 331
5.18 A description of H(A) when A ?? UrsR(Jp )\0......Page 335
5.19 Bibliographical notes\0......Page 345
6.1 Characteristic mvf?ˉs of Livsic-Brodskii nodes\0......Page 347
6.2 Connections with systems theory\0......Page 361
6.3 Bibliographical notes\0......Page 367
7 Generalized interpolation problems\0......Page 371
7.1 The Nehari problem\0......Page 372
7.2 ￥?-generating matrices\0......Page 377
7.3 Criteria for right regularity\0......Page 394
7.4 Criteria for left regularity\0......Page 403
7.5 Formulas for A when ￥? < 1\0......Page 404
7.6 The generalized Schur interpolation problem\0......Page 409
7.7 Generalized Sarason problems in S^pxq and H^pxq\0......Page 414
7.8 Resolvent matrices for the GSP in the Schur class\0......Page 417
7.9 The generalized Carath￠￥eodory interpolation problem\0......Page 421
7.10 Connections between the GSIP and the GCIP\0......Page 423
7.11 Generalized Sarason problems in ￠aC^pxp\0......Page 426
7.12 Bibliographical notes\0......Page 430
8.1 Helical extension problems\0......Page 433
8.2 Positive definite extension problems\0......Page 452
8.3 Positive definite and helical extension problems\0......Page 454
8.4 Resolvent matrices for positive definite extension problems\0......Page 456
8.5 Tangential and bitangential positive extension problems\0......Page 459
8.6 The Krein accelerant extension problem\0......Page 462
8.7 The accelerant and helical extension problems\0......Page 464
8.8 Conditions for strictly completely indeterminate HEP(g; a)\0......Page 465
8.9 Formulas for the normalized resolvent matrix for the HEP\0......Page 470
8.10 B-resolvent matrices for the accelerant case\0......Page 473
8.11 Bibliographical notes\0......Page 480
9 Darlington representations and related inverse problems\0......Page 483
9.1 D-representations of mvf?ˉs s ?? ￥D ?? S^pxq\0......Page 484
9.2 Chain scattering D-representations\0......Page 494
9.3 D-representations in the class C^pxp\0......Page 497
9.4 de Branges matrices and spectral functions\0......Page 499
9.5 The inverse 12 block problem for jpq -inner mvf?ˉs\0......Page 502
9.6 Inverse problems in special classes\0......Page 503
9.7 Jp,r -inner SI-dilations of c ?? ￥D?? C^pxp\0......Page 512
9.8 Rational approximation of mvf?ˉs of the class ￥D ?? H^pxq\0......Page 519
9.9 Bibliographical notes\0......Page 525
10.1 The matrix Muckenhoupt (A2 ) condition\0......Page 528
10.2 Criteria of strong regularity for ￥?-generating matrices\0......Page 531
10.3 Strong regularity for J-inner mvf?ˉs\0......Page 534
10.4 A mvf U ?? E ??UrsR(J1 ) that is not in L2??2\0......Page 538
10.5 Right and left strongly regular de Branges matrices\0......Page 541
10.6 LB J-nodes with characteristic mvf?ˉs in UrsR(J)\0......Page 544
10.7 Bibliographical notes\0......Page 546
11.1 Definitions of the entropy functionals\0......Page 549
11.2 Motivation for the terminology entropy functional\0......Page 550
11.3 Entropy of completely indeterminate interpolation problems\0......Page 551
11.4 Formulas for entropy functionals and their minimum\0......Page 553
11.5 A matricial generalization of the entropy inequality\0......Page 559
11.6 Bibliographical notes\0......Page 562
Bibliography\0......Page 564
Notation index\0......Page 580
Subject index\0......Page 584