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-, 1137–1138 Adele ring, 1288–1291 Bessel functions, 568–569, 1660 Adele , 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 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 , 887 C , 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-, 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, , 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 , 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 matrices, 210, 609–621 Holomorphic vector bundles, 133–134 Joint spectrum, 1182, 1185, 1826 Holomorphy, 312, 336 Jordan blocks, 1054–1055, 1057–1059 Homogeneous varieties, 1169 Julia operator, 221, 668 Homogeneous vector bundle, 143–145 Homomorphism, 1207 Howe dual pair, 1534 K H -polar decompositions, 436–441 Kalman filter, 458–459 H -procrustes problems, 436 k-Bergman space, 1117 Hypercomplex signal, 1707–1714 Kelvin inverse, 1829 Hyperholomorphic functions, 1746, 1752, Kernel , 63 1757 Kernel with negative squares, 266, 292–293, 296 Kolmogorov decomposition, 317, 664 I Koszul complex, 1014–1016, 1184 Impedance function, 781, 792 Kre˘ın accelerant extension problem, 763–765 Inclusion of operator (into a system), 784 Kre˘ın–Feller differential operator, 506 Indefinite linear algebra, 432 Kre˘ın–Feller operators, 414–415 Induced space, 325–327 Kre˘ın formula, 183–215 Inertia of solutions, 465–466 Kre˘ın’s entire operators, 572–574 Infinite dimensional linear systems, 811–831 Kre˘ın’s factorization theorem, 495 Information theoretic secrecy, 938–940 Kre˘ın space(s), 152–159, 167–181, 220–237, Inner–outer factorization, 913–917 243–250, 318, 405–406, 674, Inner product space, 1256–1258 1253–1255 Index 1857

Kre˘ın-space operators, 1278–1282 Möbius inversion, 1249 Kre˘ın space settings, 718 Module tensor product, 981 Kronecker delta, 1275, 1276 Monodromy matrix, 758 Kullback-Leibler divergence, 935 Monogenic curvelets, 1720 Monogenic function calculus, 1823–1849 L Monogenic function(s), 1340, 1349–1360, Laplace transformable signals, 882 1399–1401, 1403, 1551, 1568, 1570, Left regular representation free semigroup 1574, 1679 algebra, 1139 Monogenic signals, 1696–1698, 1701–1722 Lie algebras, 1228–1238 Monogenic wavelets, 1720 Lie groups, 1228–1238 Moore–Penrose inverses, 917–918, 929 Lifted-norm space, 637–642 Mother wavelet, 122 Limit behavior, 923–927 Multi-valued operators, 159–163 LinConnect, 138–139 Multiple kernels, 66–68 Linear canonical transforms (LCT), 870–871 Multiplication operator, 492–493 Linear connection, 130–140 Multiplier algebra, 1131–1132 Linear control systems, 882–889 Multipliers, 988–990, 1171–1172 Linear fractional transformations, 731–732 Linear momentum operator, 576–577 Linear quadratic optimal control, 454–455 N Linear relations, 160–161 Natural numbers, 1247 Linear systems and transformations, 834, 846, Nearest neighbor dynamics, 811 862–869 Negative subspace, 228 Linearization, 312 Nested codes, 944–945 l-infinity state space, 814 Neumann series expansion, 905 Link operator, 230 Nevanlinna functions, 210 Liouville type theorem, 75 Nevanlinna matrix, 212 Little multiplier theorem, 605 Nevanlinna–Pick problem, 213–214 Livsic˘ canonical system, 783–788, 790 Nevanlinna-Pick interpolation problems, 60 Local spectral function, 242, 245, 247, 248, Non-linear coherent states, 112, 117–119 252, 253 Non-smooth domains, 1447–1467 Localization operator, 114, 116 Noncommutative analytic Toeplitz algebra, Localizations of free resolutions, 1075–1076 1139–1140 Locally definitizable operators, 241–256 Noncommutative d-shift, 1138–1139 Locally invariant domain, 1237 , 1236 L-resolvent matrix, 201–204 Nonnegative operators, 204–209 L-system, 788–792, 794, 799 Normal matrices, 435–436 LU and spectral factorization, 918–923 Nullstellensatz for homogeneous ideals, Lu Qi Keng problem, 84–85 1145 Lurking isometry argument, 659, 1156 Lyapunov-Stein equation, 769, 908 O Operation on RKHS, 15–27 M Operator Martinelli kernel, 1112–1113 algebras, 1143–1145 Martinelli-Bochner integrals, 1430, 1431 boundary triplet, 183–214 Matrix differential equations, 479–483 bounded, 1812 Matrix function, 530 calculus, 1379–1385 Maximal chain of matrices, 383–384, 393 matrix completion problem, 223–225 Maxwell equations, 1439 prime, 784 McCullough-Quiggin theorem, 65–66 quaternionic linear, 1812 Minimal factorization, 290 representations, 345–368 Minimal polynomial, 971 Orthogonal polynomials, 90–91, 620 1858 Index

Overlapping space, 632 Quaternionic FT (qFT), 1665 Oversampling, 95–98 Quaternionic Hilbert spaces, 1726 Quaternionic linear operators, 1812 Quaternionic linear spaces, 1725–1741 P Quaternionic Pontryagin spaces, 1758–1765 p-adic analysis, 1287 Paley–Wiener–Levinson’s theorem, 99–100 Paley–Wiener spaces, 91–102, 475–477, 505, 567–568 R Paley–Wiener theorem, 98–99, 492 Radar ambiguity function, 847 Parfenov’s conditions, 419–421 Radial algebra, 1600–1603 Parrott completion problem, 223–225 Radial Schrödinger Operators, 577–578 Partial isometry, 229, 756–757 Random Hamiltonian perturbations, 447, 448 Perturbations, 242, 245, 250–253, 877–879 Rational dilation problem, 227 Phase function, 594–603 Rational transfer functions, 885–886 Pick interpolation problem, 59–68 Reachability operator, 906 Pick operator, 689 Real alternative algebra, 1634–1639 Pick-Nevanlinna interpolation problem, 50–51 Realization theory, 906–911, 1751, 1753, Plancherel’s formula, 476 1757–1758 Plane wave decomposition, 1836, 1839–1843 Redheffer parametrization, 698–701 (Pluri)potential theory, 76–77 Redheffer transform, 708–709 Plurisubharmonic function, 74–75 Reduction operator, 190 Poisson summation formula, 96 Regular de Branges matrix, 746 Polar codes, 946–949 Regular dilation, 1097–1105 Polynomial functions, 1114 Regular functions, 1551–1555, 1560, 1571, Polynomial spaces, 770 1573 Pontryagin space, 228, 348, 354, 362, 363, Reinhardt domain, 82 366, 368, 381–383, 386, 387, 527, Reparameterizations, 501 1255 Representation theory, 145, 1473 Pontryagin space symmetric operators, 194 Reproducing kernel(s), 127–146, 316, 317, Positive definite function, 518–519 628 Positive definite kernel, 112–113 Reproducing kernel Hilbert module, 1047 Positive operator valued (POV), 115–116 Reproducing kernel Hilbert space (RKHS), Potapov class, 729 15–20, 88–89, 113, 633–637, Potapov-Ginzburg transform, 229, 729–730 650–669, 683–684, 725–728, 1130, Potapov’s theorem, 739 1750, 1762, 1763 p-prime arithmetic-functional representation, of , 32–50 1268 Reproducing property, 4–8 p-Prime W-probability spaces, 1306–1308 Riccati equations, 769 Pseudoconvex function, 75 Riemann sphere, 893 Pullback space, 642–645, 669–671 Riemann surface(s), 771, 1239–1241 Pyatkov’s approach, 408–409 Riesz basis property, 404 Riesz-Dunford formula, 1825 Riesz-Herglotz representation, 530–531 Q Riesz-Laplace transform, 1717 Quadratic formula test, 641 Riesz representation theorem, 641 Quantitative Hartogs-Rosenthal theorems, Riesz transforms, 1714–1720, 1831 1395, 1413–1418 Right external factorization, 910 Quasi-free Hilbert modules, 987–988, Rigidity phenomenon, 1079, 1161 RL , 1816 1046–1047 S .T / Quaternion(s), 1492–1494, 1502, 1549–1575, Robust stabilization problem, 892–899 1632, 1633, 1635, 1639, 1828 Rotationally invariant frames, 1720 Quaternionic analysis, 1369–1390, 1394, 1418, Row contraction, 1020, 1129 RR , 1816 1424, 1425, 1432 S .T / Index 1859

S Split exact sequence, 1062 Sampling theorem, 87–89, 835–837, 840, 859, Split functional calculus, 1194–1201 868 Split spectrum, 1192–1193 Sarason’s theorem, 61–63 Square-root algorithm, 915–917 SC-functional calculus, 1788 S-spectrum, 1798, 1799, 1803, 1805, 1815 Scattering system, 791, 794, 800, 803 Stabilization problem, 889–891 Sce’s theorem, 1496 Stable transfer functions, 885–887 Schäffer dilation space, 979 State equivalence, 908–909 Schrödinger equations, 517–518, 623–629 State space realization, 904 Schrödinger operators, 770 State space theory, 928 Schur algorithm, 1749, 1767–1769 State transitions, 904 Schur-class(es), 214 Steerability, 1702, 1719 function, 647–650, 652 Stieltjes function, 797 interpolation theory, 687–701 Stinespring, 57 Schur functions, 1746–1750 Stinespring dilation theorem, 226 Schur parameter, 264 Stochastic realization, 457–458 Schur transformation, 265–275 String equation, 506–508, 515–517 , 1654 Strong localization property, 600 s-contractive localization property, df, 65 Strong stabilizability, 892 Secrecy codes construction, 941–949 Strong vs. weak secrecy, 953–956 Secret message agreement versus secret key Structure-preserving algorithms, 435 agreement, 952–953 Sturm–Liouville differential expression, 188 Segal-Bargmann space, 42–43 Sturm-Liouville problems, 413–414 Semi/quasi-separability, 904–906 Submodule, 1158 Semibounded spectrum, 537–538 Suita conjecture, 77–78 Sesqui-linear form, 1258–1259 Superposition principle, 1212 S-functional calculus, 1788, 1797–1804 S-Weyl relations, 1617–1622 Shannon’s sampling theorem, 92–95 Symmetric Fock space, 1138–1143 S-hermitian systems, 527 Symmetric relations, 167–172 Shift-invariant subspaces, 105–108 Synchronization, 811–831 Shor’s algorithm, 934 System Signature operator, 1255 accretive, 799 Similarity problem, 424–425 accumulative, 801 Simplicial monogenics, 1477 bi-unitarily equivalent, 795 Single contractions, 1095–1097 L-system, 790 Slice hyperholomorphic functions, 1788–1797 minimal, 794 Slice hyperholomorphicity, 1633, 1647 Livsic˘ canonical, 783–788 Slice regular functions, 1570, 1571, 1573, minimal, 784 1574 Szego,˝ 42 Słodkowski’s spectra, 1191–1193 Solenoidal and irrotational vector field, 1433 Spatial invariance, 811–831 T Special affine transformation, 869 T -admissible open set, 1816 Spectral factorization, 459–461, 918 Taylor and Laurent series, 1533–1536 Spectral functions, 754–756, 762–763 Taylor functional calculus, 1201–1213 Spectral mapping property, 1189–1190, Taylor invertibility, 1014–1016 1211–1212 Taylor regular, 1184, 1185 Spectral mapping theorem, 1842 Taylor spectrum, 1182–1191 Spectrum, 241–256, 1824 Tensor product topological space, 1265–1267 Spherical contraction, 1109 Time-harmonic electromagnetic fields, Spherical dilations, 1108–1111 1439–1444 Spherical isometry, 1109 Titchmarsh-Weyl coeffcients, 526, 530–531 Spin group approach, 1667 Toeplitz algebra, 1143–1144 Spiral phase quadrature transform, 1716 Toeplitz kernel approach, 603–605 1860 Index

Toeplitz operator, 121, 645 Von Neumann inequality, 982 characterization, 684–685 Von Neumann type inequality, 1107–1108 equation, 1002–1004 Von Neumann’s inequality, 1147 Transfer function, 785, 791 V.P. Potapov’s method, 692 Trigonometric polynomials, 89–90 Two-dimensional Hamiltonian system, 525–543 W Wandering subspace, 996 Wavelets, 122–124, 1673–1698 U Weak corona property, 1016–1017 Unbounded operators, 1221–1241 Weighted multishift, 1109 Undersampling, 95–101 Wet paper coding, 960–962 Uniform approximation by monogenic Weyl calculus, 1837–1839 functions, 1413 Weyl coefficient, 503 Unitary boundary pairs/triplets, 185, 190 Weyl functions, 183–215, 378, 385, 387–389 Unitary dilation, 221 Wiener algebra, 887 Unitary invariant, 994 Wigner distribution function, 847–859, 867 Unitary representations, 1228–1230 Windowed Fourier transform, 844–845 Witt basis, 1584 W*-probability space, 1272 V Vector coherent states (VCS), 114, 119–120 Volkmer’s approach, 409–410 Z Volterra nodes, 742–743 Zaremba decomposition or expansion, 10, 75 Von Neumann algebras, 1267–1268, Zero-dimensional indices, 903 1272–1282, 1296–1297, 1303–1314 z-transform theory, 902–903