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A Abstract Factorization Approach, 887–889 Abstract Interpolation Index A Bergman-Hodge type decomposition, Abstract factorization approach, 887–889 1386–1387 Abstract interpolation problem, 693 Bergman kernel, 73–85 Abstract operator adjoints, 905 Bergman projection, 74 Accretive L-system, 799 Bergman space, 39–40, 1116 Accumulative system, 801 Besov-Sobolev space, 1137–1138 Adele ring, 1288–1291 Bessel functions, 568–569, 1660 Adele von Neumann algebra, 1308–1310 Beurling type theorems, 1160–1161 Adele W-probability space, 1328–1333 Beurling–Lax–Halmos theorem, 998 Admissible kernel, 135 Beurling–Malliavin majorant, 583 Algebraic analysis, 1555, 1603 Beurling–Malliavin multiplier theorem, Algebraic isomorphism, 1168–1169 582–591 Algebraic Riccati equations, 441–445, Beurling’s representations, 1059–1061 463–467 Bezout equation, 460–461, 891 Ambiguity transformation, 847, 859–861 Biholomorphic mappings, 80–83 Ando dilation, 1106–1108 Birth-and-death processes, 519 Andô’s theorem, 225, 982 Blaschke condition, 494–495 Antisymmetric Fock space, 1142 Blecher-Effros-Ruan axioms, 226 Arithmetic algebra, 1251, 1297–1298 Block-column operator matrix, 662 Arithmetic-function(s), 1246–1282 Block-tridiagonal system, 921–923 tensor(ing), 1264–1269 BL.V /, 1814 tensor product algebra, 1265 Bochner-Martinelli formula, 1826 Arithmetic p-prime probability space, Bognár-Krámli factorization, 230–232 1262 Borichev’s construction, 585–588 Arveson space, 1778, 1781 Boundary triplets, 183–215 Associated pairs, 733–734 Boundary value(s), 1222 Asymptotic behavior, 345–368 Boundary value problems, 1370–1390 Atomization procedure, 601–603 Bounded real lemma, 455–456 Azukawa pseudometric function, 78 Brangesian complementary space, 647 Brehmer conditions, 1105 BR.V /, 1814 B B.V /, 1814 Backward multishift, 1098 Backward shift, 1095 Banach algebra, 887 C Banach space, 875–879, 1183 Canonical coherent states (CCS), 116–117 Bandlimited functions, 835, 838–842 Canonical (co-prime) external forms, 910 Beals’ approach, 401–402 Canonical factorizations, 928 Berezin transform, 121, 1017 Canonical forms, 433–435, 907–908 © Springer Basel 2015 1853 D. Alpay (ed.), Operator Theory, DOI 10.1007/978-3-0348-0667-1 1854 Index Canonical functional models, 673–674 Conformally invariant differential operators, Canonical system, 374–380, 501, 527–533, 1477, 1481–1484, 1512, 1525 536–543, 611–613, 616, 623–629, Conjectures, 1165–1167 771 Constrained dilations, 1150–1151 Capacity approaching code, 944 Continuous wavelet transform, 123, 845 Carleson measures, 1170–1171 Contractions, 1118 Cartan domain, 1118 Contractive Hilbert modules, 978, 991–996 Cartwright class, 535 Contractive module map, 236 Cauchy formula, 1358–1360, 1642, 1644 Convolution product, 1661 Cauchy kernel, 1827, 1835–1837 Coprime factorization, 890–891, 899 Cauchy transform, 1449, 1452–1459, 1464 Corona condition, 1043 Cauchy-Kovalevskaya (CK-) extension, Corona theorem, 1173–1175 1593–1594, 1623–1625, 1680 Coupling method, 200–201 Cauchy-Kowaleski extension, 1836 Cowen-Douglas bundles, 128–129 Cauchy-Pompeiu integral representation Cowen-Douglas class, 1037–1046 formulas, 1404, 1411–1412 Cowen-Douglas Hilbert modules, 985–987 Cauchy-Riemann equations, 1831–1834 Crossover probability, 937 Cauchy-Riemann operator, 1349–1352, 1357, Cryptography, 933–934 1370, 1380 Curvature equality, 1041–1043 Chain Koszul complex, 1190 Curvature inequality, 1024–1026 Chern covariant derivative, 133 Curvature invariant, 1161–1164 Chordal distance, 893–894 Class N.R/, 793 D N0.R/, 795–796 d-Contraction, 1129 N1.R/, 796 de Branges Hilbert spaces, 559–569 N01.R/, 796 de Branges matrix, 745–746 S.R/, 799 de Branges spaces, 376, 380, 382, 386, 389, S ˛ , 800 490–521, 609–621, 626, 770 S0.R/, 799 de Branges’ inclusion theorems, 740–742 S 1.R/, 801 de Branges–Rovnyak spaces, 47–50, 632–676, 1 S0 .R/, 801 682–701, 770–771, 1781 Classifying quasimorphism, 136 Decoding procedures, 940–941 Clifford algebra, 1341–1349, 1495, 1498, Defect spaces, 221 1501, 1550, 1567, 1571, 1632, 1633, Deficiency indices, 1235 1636, 1646, 1827–1831 Definitizable operators, 405–406 Clifford analysis, 1447–1467, 1473–1477, Definitizable relations, 172 1526, 1529–1531, 1651–1670, Diagonal Hamiltonian, 540–541 1673–1698, 1831–1843 Dichotomy, 918 Clifford-Hermite basis, 1654 Differential equations, 169, 177 Closed symmetric operators, 552–553 Dilation positive definite kernels, 53 Cochain Koszul complex, 1190 Dilation theory, 978–980, 1093–1122 Coding procedures, 940–941 Dirac operator, 1654 Coefficient of dynamic compliance, 507 Dirac-Kre˘ın (DK) systems, 765–767 Coherent states, 113, 129 Direct spectral problem, 760–762 Colligation matrix, 653, 656 Dirichlet boundary conditions, 625, 629 Combinatorial free probability theory, Dirichlet characters, 1248 1248 Dirichlet L-function, 1247 Commutant lifting theorem, 220–237, 1063 Dirty paper coding. See Wet paper coding Commutative diagrams, 234 Discrete boundary values, 1625–1627 Commuting isometries, 1095 Discrete Cauchy integral formula, 1614–1617 Complete nonselfadjointness, 555–556 Discrete Clifford analysis, 1610–1628 Complete Pick property, 1154–1156 Discrete Dirac operators, 1612–1614 Conformal monogenic signal, 1722 Discrete time systems, 884–885 Index 1855 Disk algebra, 886 Free random variables, 1249 Doubly infinite operator, 903 Fueter series, 1751, 1754, 1776–1782 Drury–Arveson, 984 Fueter’s theorem, 1491–1505 module, 1021 Fueter-Sce mapping theorem, 1788, 1789, space, 1127 1793 d-Shift, 1132, 1141 Full range completeness, 401–402 d-Shift among d-contractions, 1148–1149 Function d-Shift among row contractions, 1145–1146 Herglotz-Nevanlinna, 792 D-space, 1111, 1112 impedance, 792 d-tuples, 1129 left monogenic, 1834 right monogenic, 1833 theory, 1472 E transfer, 785 Eigenfunction approach, 1655–1663 of an L-system, 791 Eigenfunction expansions, 484–486 Functional space, 569–574 Eigenvalues, 538 Fundamental solution, 1487–1488 EndL.V /,EndR.V /, 1813 Fundamental symmetry, 228 Erasure probability, 937 Essential Taylor spectrum, 1194 Euclidean Cauchy-Riemann operators, G 1401–1403 Gauss-Bonnet theorem, 1163–1164 Euclidean Dirac operators, 1399–1401 Gelfan-Tsetlin basis, 1536–1540 Euler characteristic, 1162–1163 Gelfand-Levitan equations, 628 Euler subalgebra, 1321–1324 Generalized canonical models, 1037–1046, Euler totient function, 1286, 1298–1301 1068–1071 Evaluation map, 974 Generalized Cayley transform, 554–555 Eve’s channel, 949 Generalized Fourier transforms, 757–760 Exponential growth, 512–513 Generalized Nevanlinna functions, 346–368 ()-extension(s), 789, 791, 801 Generalized poles and zeros, 356–361 Extension theory, 177 Generalized roots of –1 approach, 1652–1667 External factorizations, 928 Generalized Schur functions, 263, 264, 266–267, 1769–1772 Geometric Fourier transform (GFT), F 1663–1667 Factor analysis, 929 GNS theorem, 57 F-functional calculus, 1788, 1808–1812 Gram matrix, 987 Finite (block-)matrices, 906 Grassmann manifold, 128 Finite dimensional indefinite inner product Grauert’s theorem, 986 spaces, 431–448 Group von Neumann algebras, 1274–1275 Finite spectral measure, 540 Growth functions, 508–511 Fischer decomposition, 1595–1597, Gyroscopic systems, 445–448 1617–1622 Fractional Fourier transform, 862–869 Fractional powers of Laplacian, 1492, 1501 H Frames, 1695–1698 Haar-measure space, 1291 Fredholm index, 924 Half Range Completeness, 399–404 Fredholm kernel, 927 Hamburger moment problem, 210–215, Fredholm operator, 875–879 477–479, 574–576 Free cumualants, 1273 Hamburger-Nevanlinna theorem, 210 Free distributional data, 1269–1272 Hamiltonian eigenvalue problem, 441 Free moments, 1249 Hamiltonian matrices, 441–448 Free probabilistic model, 1247 Hamiltonian Schur form, 441–445 Free probability, 1249–1251, 1293–1296 Hamiltonian system, 528 Free product algebra, 1251 Hankel geometry, 912–913 1856 Index Hardy algebra, 886, 897 Inner resolutions, 1073–1075 Hardy space, 925, 973, 1748, 1750, Integral representation formulas with 1763–1766, 1773 remainders, 1409 Harmonic analysis, 1236, 1846–1848 Integral transforms, 1360–1364 2 Hm Integrality of the curvature invariant, 1164 df, 66 Interpolating sequence, 1172–1173 H 1-control, 456–457 Interpolation, 279–287 HELP-type inequality, 412–413 problems, 701–708 Hereditary functional calculus, 1049–1053 Intertwining operators, 237 Herglotz’s theorem, 60 Invariant lagrangian subspaces, 461–463 Herglotz–Nevanlinna function, 792–796, 798 Invariant subspaces, 1158 Hermite-Biehler function, 511–512 Inverse monodromy problem, 767–769 Hermitian Dirac operators, 1589, 1603–1606 Inverse spectral problem, 760–762 Hermitian kernel, 314–318 Inverse spectral theory, 626–629 Hermitian matrix, 1824 Inverse Stieltjes function, 800 Hermitian monogenic functions, 1590, 1599 Invertible matrix, 433 Hermitian structure, 132–134 Invertible multipliers, 1069–1071 H-holomorphic functions, 1379–1381 Isometric dilation, 221 H 1-functional calculus, 1492, 1501–1505 Isometric isomorphism, 1167–1168 Hilbert function spaces, 651, 1128, 1130, 1131 Isometric module maps, 1007 Hilbert module approach, 969–1026, 1127, Isometric multipliers, 1072–1073 1130 Isometric relations, 172–181 Hilbert spaces, 102, 187–192, 220–227, 473– Isomorphic Kre˘ın-space, 1263 487, 534–535, 905, 1094–1122, Isometric and unitary operators, 910–911 1223, 1225, 1227–1230 Hilbert transform, 1703–1707 Hilbert–Samuel polynomial, 1007–1010 J Hilbert–Schmidt space, 925 J -Lagrangian, 453 Hilbert–Schmidth cross commmutator, 1087 J -unitary matrix polynomial, 263, 287 Holomorphic functions, 983–991 J.von Neumann approach, 193 in two complex variables, 1426–1433 Jacobi
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