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Linearly similar Szökefalvi-Nagy-Foias model in a domain


Author: D. V. Yakubovich
Translated by: V. V. Kapustin
Original publication: Algebra i Analiz, tom 15 (2003), nomer 2.
Journal: St. Petersburg Math. J. 15 (2004), 289-321
MSC (2000): Primary 47A45
Published electronically: January 30, 2004
MathSciNet review: 2052133
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  • 1. D. Z. Arov, Passive linear steady-state dynamical systems, Sibirsk. Mat. Zh. 20 (1979), no. 2, 211–228, 457 (Russian). MR 530486
  • 2. D. Z. Arov and M. A. Nudel′man, Conditions for the similarity of all minimal passive realizations of a given transfer function (scattering and resistance matrices), Mat. Sb. 193 (2002), no. 6, 3–24 (Russian, with Russian summary); English transl., Sb. Math. 193 (2002), no. 5-6, 791–810. MR 1957950, 10.1070/SM2002v193n06ABEH000657
  • 3. V. Vasyunin and S. Kupin, Criteria for the similarity of a dissipative integral operator to a normal operator, Algebra i Analiz 13 (2001), no. 3, 65–104 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 13 (2002), no. 3, 389–416. MR 1850188
  • 4. I. C. Gohberg and M. G. Kreĭn, On a description of contraction operators similar to unitary ones, Funkcional. Anal. i Priložen. 1 (1967), 38–60 (Russian). MR 0213907
  • 5. G. M. Gubreev, Spectral theory of regular quasi-exponentials and regular 𝐵-representable vector functions (the projection method: 20 years later), Algebra i Analiz 12 (2000), no. 6, 1–97 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 12 (2001), no. 6, 875–947. MR 1816512
  • 6. V. V. Kapustin, Spectral analysis of almost unitary operators, Algebra i Analiz 13 (2001), no. 5, 44–68 (Russian); English transl., St. Petersburg Math. J. 13 (2002), no. 5, 739–756. MR 1882863
  • 7. A. L. Lihtarnikov and V. A. Jakubovič, A frequency theorem for equations of evolution type, Sibirsk. Mat. Ž. 17 (1976), no. 5, 1069–1085, 1198 (Russian). MR 0430454
  • 8. M. M. Malamud, A criterion for a closed operator to be similar to a selfadjoint operator, Ukrain. Mat. Zh. 37 (1985), no. 1, 49–56, 134 (Russian). MR 780914
  • 9. M. M. Malamud, On the similarity of a triangular operator to a diagonal operator, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 270 (2000), no. Issled. po Linein. Oper. i Teor. Funkts. 28, 201–241, 367 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N. Y.) 115 (2003), no. 2, 2199–2222. MR 1795645, 10.1023/A:1022888921572
  • 10. S. N. Naboko, Functional model of perturbation theory and its applications to scattering theory, Trudy Mat. Inst. Steklov. 147 (1980), 86–114, 203 (Russian). Boundary value problems of mathematical physics, 10. MR 573902
  • 11. S. N. Naboko, Conditions for similarity to unitary and selfadjoint operators, Funktsional. Anal. i Prilozhen. 18 (1984), no. 1, 16–27 (Russian). MR 739086
  • 12. S. N. Naboko and M. M. Faddeev, Operators of the Friedrichs model that are similar to a selfadjoint operator, Vestnik Leningrad. Univ. Fiz. Khim. vyp. 4 (1990), 78–82, 114 (Russian, with English summary). MR 1127406
  • 13. N. K. Nikol′skiĭ, Lektsii ob operatore sdviga, “Nauka”, Moscow, 1980 (Russian). MR 575166
    N. K. Nikol′skiĭ, Treatise on the shift operator, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273, Springer-Verlag, Berlin, 1986. Spectral function theory; With an appendix by S. V. Hruščev [S. V. Khrushchëv] and V. V. Peller; Translated from the Russian by Jaak Peetre. MR 827223
  • 14. B. S. Pavlov, Conditions for separation of the spectral components of a dissipative operator, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 123–148, 240 (Russian). MR 0365199
  • 15. B. Pavlov, Nonphysical sheet for perturbed Jacobian matrices, Algebra i Analiz 6 (1994), no. 3, 185–199; English transl., St. Petersburg Math. J. 6 (1995), no. 3, 619–633. MR 1301837
  • 16. I. I. Privalov, Graničnye svoĭstva analitičeskih funkciĭ, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1950 (Russian). 2d ed.]. MR 0047765
  • 17. V. A. Ryzhov, Absolutely continuous and singular subspaces of a nonselfadjoint operator, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222 (1995), no. Issled. po Linein. Oper. i Teor. Funktsii. 23, 163–202, 309 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 87 (1997), no. 5, 3886–3911. MR 1359998, 10.1007/BF02355830
  • 18. L. A. Sahnovič, Dissipative operators with absolutely continuous spectrum, Trudy Moskov. Mat. Obšč. 19 (1968), 211–270 (Russian). MR 0250109
  • 19. B. M. Solomyak, A functional model for dissipative operators. A coordinate-free approach, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 178 (1989), no. Issled. Linein. Oper. Teorii Funktsii. 18, 57–91, 184–185 (Russian, with English summary); English transl., J. Soviet Math. 61 (1992), no. 2, 1981–2002. MR 1037765, 10.1007/BF01095663
  • 20. Béla Sz.-Nagy and Ciprian Foiaș, Harmonic analysis of operators on Hilbert space, Translated from the French and revised, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. MR 0275190
  • 21. S. I. Fedorov, On harmonic analysis in a multiply connected domain and character-automorphic Hardy spaces, Algebra i Analiz 9 (1997), no. 2, 192–240 (Russian); English transl., St. Petersburg Math. J. 9 (1998), no. 2, 339–378. MR 1468551
  • 22. A. V. Štraus, Characteristic functions of linear operators, Izv. Akad. Nauk SSSR Ser. Mat. 24 (1960), 43–74 (Russian). MR 0140950
  • 23. Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR 0089373
  • 24. V. A. Jakubovič, The frequency theorem for the case in which the state space and the control space are Hilbert spaces, and its application in certain problems in the synthesis of optimal control. II, Sibirsk. Mat. Ž. 16 (1975), no. 5, 1081–1102, 1132 (Russian). MR 0473979
  • 25. Daniel Alpay and Victor Vinnikov, Finite dimensional de Branges spaces on Riemann surfaces, J. Funct. Anal. 189 (2002), no. 2, 283–324. MR 1891851, 10.1006/jfan.2000.3623
  • 26. D. Z. Arov and M. A. Nudelman, Passive linear stationary dynamical scattering systems with continuous time, Integral Equations Operator Theory 24 (1996), no. 1, 1–45. MR 1366539, 10.1007/BF01195483
  • 27. Z. D. Arova, On 𝐽-unitary nodes with strongly regular 𝐽-inner characteristic functions in the Hardy class 𝐻^{𝑛×𝑛}₂, Operator theoretical methods (Timişoara, 1998) Theta Found., Bucharest, 2000, pp. 29–38. MR 1770312
  • 28. Sergei A. Avdonin and Sergei A. Ivanov, Families of exponentials, Cambridge University Press, Cambridge, 1995. The method of moments in controllability problems for distributed parameter systems; Translated from the Russian and revised by the authors. MR 1366650
  • 29. Joseph A. Ball, A lifting theorem for operator models of finite rank on multiply-connected domains, J. Operator Theory 1 (1979), no. 1, 3–25. MR 526287
  • 30. Joseph A. Ball and Nir Cohen, de Branges-Rovnyak operator models and systems theory: a survey, Topics in matrix and operator theory (Rotterdam, 1989) Oper. Theory Adv. Appl., vol. 50, Birkhäuser, Basel, 1991, pp. 93–136. MR 1115026, 10.1007/978-3-0348-5672-0_5
  • 31. Nour-Eddine Benamara and Nikolai Nikolski, Resolvent tests for similarity to a normal operator, Proc. London Math. Soc. (3) 78 (1999), no. 3, 585–626. MR 1674839, 10.1112/S0024611599001756
  • 32. Louis de Branges and James Rovnyak, Canonical models in quantum scattering theory, Perturbation Theory and its Applications in Quantum Mechanics (Proc. Adv. Sem. Math. Res. Center, U.S. Army, Theoret. Chem. Inst., Univ. of Wisconsin, Madison, Wis., 1965) Wiley, New York, 1966, pp. 295–392. MR 0244795
  • 33. Douglas N. Clark, On a similarity theory for rational Toeplitz operators, J. Reine Angew. Math. 320 (1980), 6–31. MR 592139, 10.1515/crll.1980.320.6
  • 34. Ruth F. Curtain and Hans Zwart, An introduction to infinite-dimensional linear systems theory, Texts in Applied Mathematics, vol. 21, Springer-Verlag, New York, 1995. MR 1351248
  • 35. R. Datko, Extending a theorem of A. M. Lyapunov to Hilbert space, J. Math. Anal. Appl. 32 (1970), 610-616. MR 2:3614
  • 36. Peter L. Duren, Theory of 𝐻^{𝑝} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
  • 37. Harry Dym, On Riccati equations and reproducing kernel spaces, Recent advances in operator theory (Groningen, 1998) Oper. Theory Adv. Appl., vol. 124, Birkhäuser, Basel, 2001, pp. 189–215. MR 1839837
  • 38. Ciprian Foias and Arthur E. Frazho, The commutant lifting approach to interpolation problems, Operator Theory: Advances and Applications, vol. 44, Birkhäuser Verlag, Basel, 1990. MR 1120546
  • 39. Paul A. Fuhrmann, Linear systems and operators in Hilbert space, McGraw-Hill International Book Co., New York, 1981. MR 629828
  • 40. Paul A. Fuhrmann, A polynomial approach to linear algebra, Universitext, Springer-Verlag, New York, 1996. MR 1393938
  • 41. Morisuke Hasumi, Invariant subspace theorems for finite Riemann surfaces, Canad. J. Math. 18 (1966), 240–255. MR 0190790
  • 42. Birgit Jacob and Jonathan R. Partington, The Weiss conjecture on admissibility of observation operators for contraction semigroups, Integral Equations Operator Theory 40 (2001), no. 2, 231–243. MR 1831828, 10.1007/BF01301467
  • 43. B. Jacob, J. R. Partington, and S. Pott, Conditions for admissibility of observation operators and boundedness of Hankel operators (to appear).
  • 44. M. A. Kaashoek and S. M. Verduyn Lunel, Characteristic matrices and spectral properties of evolutionary systems, Trans. Amer. Math. Soc. 334 (1992), no. 2, 479–517. MR 1155350, 10.1090/S0002-9947-1992-1155350-0
  • 45. N. J. Kalton and C. Le Merdy, Solution of a problem of Peller concerning similarity, J. Operator Theory 47 (2002), no. 2, 379–387. MR 1911852
  • 46. V. Komornik, Exact controllability and stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. The multiplier method. MR 1359765
  • 47. Stanislav Kupin, Linear resolvent growth test for similarity of a weak contraction to a normal operator, Ark. Mat. 39 (2001), no. 1, 95–119. MR 1821084, 10.1007/BF02388793
  • 48. S. Kupin and S. Treil, Linear resolvent growth of a weak contraction does not imply its similarity to a normal operator, Illinois J. Math. 45 (2001), no. 1, 229–242. MR 1849996
  • 49. Christian Le Merdy, On dilation theory for 𝑐₀-semigroups on Hilbert space, Indiana Univ. Math. J. 45 (1996), no. 4, 945–959. MR 1444474, 10.1512/iumj.1996.45.1191
  • 50. Christian Le Merdy, The similarity problem for bounded analytic semigroups on Hilbert space, Semigroup Forum 56 (1998), no. 2, 205–224. MR 1490293, 10.1007/PL00005942
  • 51. -, The Weiss conjecture for bounded analytic semigroups (to appear).
  • 52. Christian Le Merdy, A bounded compact semigroup on Hilbert space not similar to a contraction one, Semigroups of operators: theory and applications (Newport Beach, CA, 1998), Progr. Nonlinear Differential Equations Appl., vol. 42, Birkhäuser, Basel, 2000, pp. 213–216. MR 1788884
  • 53. -, Similarities of $\omega $-accretive operators, Prépubl. Lab. Math. Besançon no. 2002/07, Univ. France Comté, Centre Nat. Rech. Sci., Besançon, 2002.
  • 54. J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 8, Masson, Paris, 1988 (French). Contr\cflex olabilité exacte. [Exact controllability]; With appendices by E. Zuazua, C. Bardos, G. Lebeau and J. Rauch. MR 953547
    J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 2, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 9, Masson, Paris, 1988 (French). Perturbations. [Perturbations]. MR 963060
  • 55. M. S. Livšic, N. Kravitsky, A. S. Markus, and V. Vinnikov, Theory of commuting nonselfadjoint operators, Mathematics and its Applications, vol. 332, Kluwer Academic Publishers Group, Dordrecht, 1995. MR 1347918
  • 56. Sjoerd M. Verduyn Lunel and Dmitry V. Yakubovich, A functional model approach to linear neutral functional-differential equations, Integral Equations Operator Theory 27 (1997), no. 3, 347–378. MR 1433007, 10.1007/BF01324734
  • 57. Alan McIntosh, Operators which have an 𝐻_{∞} functional calculus, Miniconference on operator theory and partial differential equations (North Ryde, 1986) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 14, Austral. Nat. Univ., Canberra, 1986, pp. 210–231. MR 912940
  • 58. Scott McCullough, Commutant lifting on a two holed domain, Integral Equations Operator Theory 35 (1999), no. 1, 65–84. MR 1707931, 10.1007/BF01225528
  • 59. N. K. Nikolski, Contrôlabilité à une renormalisation près et petits opérateurs de contrôle, J. Math. Pures Appl. (9) 77 (1998), no. 5, 439–479 (French, with English summary). MR 1626796, 10.1016/S0021-7824(98)80027-5
  • 60. Nikolai Nikolski and Vasily Vasyunin, Elements of spectral theory in terms of the free function model. I. Basic constructions, Holomorphic spaces (Berkeley, CA, 1995) Math. Sci. Res. Inst. Publ., vol. 33, Cambridge Univ. Press, Cambridge, 1998, pp. 211–302. MR 1630652
  • 61. Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz; Translated from the French by Andreas Hartmann. MR 1864396
    Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 2, Mathematical Surveys and Monographs, vol. 93, American Mathematical Society, Providence, RI, 2002. Model operators and systems; Translated from the French by Andreas Hartmann and revised by the author. MR 1892647
  • 62. Vern I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR 868472
  • 63. A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
  • 64. Gilles Pisier, A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc. 10 (1997), no. 2, 351–369. MR 1415321, 10.1090/S0894-0347-97-00227-0
  • 65. A. J. Pritchard and D. Salamon, The linear quadratic control problem for infinite-dimensional systems with unbounded input and output operators, SIAM J. Control Optim. 25 (1987), no. 1, 121–144. MR 872455, 10.1137/0325009
  • 66. Olof J. Staffans, Quadratic optimal control of well-posed linear systems, SIAM J. Control Optim. 37 (1999), no. 1, 131–164 (electronic). MR 1645436, 10.1137/S0363012996314257
  • 67. Olof Staffans and George Weiss, Transfer functions of regular linear systems. II. The system operator and the Lax-Phillips semigroup, Trans. Amer. Math. Soc. 354 (2002), no. 8, 3229–3262 (electronic). MR 1897398, 10.1090/S0002-9947-02-02976-8
  • 68. N. G. Makarov and V. I. Vasjunin, A model for noncontractions and stability of the continuous spectrum, Complex analysis and spectral theory (Leningrad, 1979/1980) Lecture Notes in Math., vol. 864, Springer, Berlin-New York, 1981, pp. 365–412. MR 643386
  • 69. Michael Voichick, Invariant subspaces on Riemann surfaces, Canad. J. Math. 18 (1966), 399–403. MR 0190791
  • 70. George Weiss, Admissible observation operators for linear semigroups, Israel J. Math. 65 (1989), no. 1, 17–43. MR 994732, 10.1007/BF02788172
  • 71. George Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim. 27 (1989), no. 3, 527–545. MR 993285, 10.1137/0327028
  • 72. Daoxing Xia, On the kernels associated with a class of hyponormal operators, Integral Equations Operator Theory 6 (1983), no. 3, 444–452. MR 701029, 10.1007/BF01691907
  • 73. D. V. Yakubovich, Spectral properties of smooth finite rank perturbations of a planar Lebesgue spectrum cyclic normal operator, Report no. 15 (1990/91), Inst. Mittag-Leffler, Stockholm.
  • 74. Dmitry V. Yakubovich, Spectral properties of smooth perturbations of normal operators with planar Lebesgue spectrum, Indiana Univ. Math. J. 42 (1993), no. 1, 55–83. MR 1218707, 10.1512/iumj.1993.42.42005
  • 75. D. V. Yakubovich, Dual piecewise analytic bundle shift models of linear operators, J. Funct. Anal. 136 (1996), no. 2, 294–330. MR 1380657, 10.1006/jfan.1996.0032
  • 76. -, A similarity version of the Nagy-Foias model, duality and exact controllability, 17th International Conference on Operator Theory (Timisoara, 1998): Abstracts of Reports.
  • 77. Dmitry V. Yakubovich, Subnormal operators of finite type. II. Structure theorems, Rev. Mat. Iberoamericana 14 (1998), no. 3, 623–681. MR 1681587, 10.4171/RMI/247
  • 78. Enrique Zuazua, Some problems and results on the controllability of partial differential equations, European Congress of Mathematics, Vol. II (Budapest, 1996) Progr. Math., vol. 169, Birkhäuser, Basel, 1998, pp. 276–311. MR 1645833
  • 79. A. S. Tikhonov, A functional model and duality of spectral components for operators with a continuous spectrum on a curve, Algebra i Analiz 14 (2002), no. 4, 158–195 (Russian); English transl., St. Petersburg Math. J. 14 (2003), no. 4, 655–682. MR 1935922
  • 80. Piotr Grabowski and Frank M. Callier, Admissible observation operators. Semigroup criteria of admissibility, Integral Equations Operator Theory 25 (1996), no. 2, 182–198. MR 1388679, 10.1007/BF01308629

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Additional Information

D. V. Yakubovich
Affiliation: St. Petersburg State University, Russia, and Autonomous University of Madrid, Spain
Email: dmitry.yakubovich@nam.es

DOI: https://doi.org/10.1090/S1061-0022-04-00811-8
Keywords: Contraction, dissipative operator, generalized characteristic function, semigroup, functional calculus
Received by editor(s): August 20, 2002
Published electronically: January 30, 2004
Article copyright: © Copyright 2004 American Mathematical Society