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Evolution of the concept of the characteristic function of a linear operator

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Abstract

This is a brief survey of the development and applications of the concept of the characteristic function for different classes of linear operators.

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References

  1. M. S. Livshits, “On a class of linear operators in the Hilbert space,”Mat. Sb.,19, 239–260 (1946).

    Google Scholar 

  2. M. S. Livshits, “Isometric operators with equal deficiency numbers; quasiunitary operators,”Mat. Sb.,26, 247–264 (1950).

    Google Scholar 

  3. M. S. Livshits and V. P. Potapov, “Multiplication theorem for the characteristic matrix functions,”Dokl. Akad. Nauk SSSR,72, No. 4, 625–628 (1950).

    Google Scholar 

  4. Yu. L. Shmul'yan, “Operators with a degenerate characteristic function,”Dokl. Akad. Nauk SSSR,93, No. 6, 986–988 (1953).

    Google Scholar 

  5. V. T. Polyatskii, “On the reduction of quasiunitary operators to triangular form,”Dokl. Akad. Nauk SSSR,113, No. 4, 756–759 (1957).

    Google Scholar 

  6. M. S. Livshits, “On the spectral decomposition of linear operators which are not self-adjoint,”Mat. Sb.,34, 144–199 (1954).

    Google Scholar 

  7. B. Szökefalvy-Nagy and C. Foias, “Models fonctionelles des contractions de l'espace de Hilbert. La fonction caracteristique,”Comptes Rend.,256, 3236–3239 (1963).

    Google Scholar 

  8. B. Szökefalvy-Nagy and C. Foias, “Propriétés des fonctions caractéristiques, modèles triangulaes et une classification des contractions,”Comptes Rend.,258, 3413–3415 (1963).

    Google Scholar 

  9. B. Szökefalvy-Nagy and C. Foias,Harmonic Analysis of the Operators in Hilbert Space [Russian translation], Mir, Moscow (1970).

    Google Scholar 

  10. A. V. Kuzhel', “Multiplication theorem for the characteristic matrix functions of nonunitary operators,”Nauch. Dokl. Vyssh. Shk., Ser. Fiz.-Mat., No. 3, 33–41 (1959).

    Google Scholar 

  11. A. V. Kuzhel', “The characteristic operator function of an arbitrary bounded operator,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 3, 233–236 (1968).

    Google Scholar 

  12. A. V. Kuzhel', “The generalization of the Sz.-Nagy-Foias theorem on factorization of a characteristic operator function,”Acta Sci. Math.,30, 225–234 (1969).

    Google Scholar 

  13. Ch. Devis and C. Foias, “Operators with bounded characteristic function and theirJ-unitary dilation,”Acta Sci. Math.,32, 127–139 (1971).

    Google Scholar 

  14. L. A. Sakhnovich, “Dissipative operators with absolutely continuous spectrum,”Tr. Mosk. Mat. Obshch.,19, 211–270 (1968).

    Google Scholar 

  15. D. Clark, “On models for noncontractions,”Acta Sci. Math.,36, 5–6 (1974).

    Google Scholar 

  16. B. McEnnis, “Purely contractive analytic functions and characteristic functions of noncontractions,”Acta Sci. Math.,41, 161–172 (1979).

    Google Scholar 

  17. B. McEnnis, “Characteristic functions and dilatations of noncontractions,”J. Operator Theory,3, 71–87 (1980).

    Google Scholar 

  18. B. McEnnis, “Models for operators with bounded characteristic functions,”Acta Sci. Math.,43, 71–90 (1981).

    Google Scholar 

  19. N. G. Makarov, “Stability of the essential spectrum of operators close to unitary operators,”Funkts. Anal. Prilozh.,16, No. 3, 72–73 (1982).

    Google Scholar 

  20. V. M. Brodskii, I. Ts. Gokhberg, and M. G. Krein, “Definition and principal properties of the characteristic function of aJ-knot,”Funkts. Anal. Prilozh.,4, No. 1, 88–90 (1970).

    Google Scholar 

  21. V. M. Brodskii, “Theorems of multiplication and division for the characteristic functions of an invertible operator,”Acta Sci. Math.,32, 165–175 (1971).

    Google Scholar 

  22. V. N. Polyakov, “On the theory of characteristic functions of linear operators,”Izv. Vyssh. Uchebn. Zaved., Ser. Mat.,63, No. 8, 53–59 (1967).

    Google Scholar 

  23. V. N. Polyakov, “One problem in the theory of characteristic functions of linear operators,”Mat. Zametki,9, No. 2, 171–180 (1971).

    Google Scholar 

  24. V. N. Polyakov, “Multiplication theorem for characteristic functions of linear operators,”Izv. Vyssh. Uchebn. Zaved., Ser. Mat.,132, No. 5, 73–76 (1973).

    Google Scholar 

  25. V. M. Adamyan and D. Z. Arov, “On a class of scattering operators and on characteristic operator functions of contractions,”Dokl. Akad. Nauk SSSR,160, No. 1, 9–12 (1965).

    Google Scholar 

  26. V. M. Adamyan and D. Z. Arov “On scattering operators and semigroups of contractions in Hilbert spaces,”Dokl. Akad. Nauk SSSR,165, No. 1, 9–12 (1965).

    Google Scholar 

  27. V. M. Adamyan and D. Z. Arov, “On unitary couplings of semiunitary operators,”Mat. Issledovaniya (Kishinev),1, No. 2, 3–64 (1966).

    Google Scholar 

  28. V. M. Adamyan and D. Z. Arov, “The general solution to the problem of linear prediction of stationary processes,”Teor. Veroyatn. Primen.,13, No. 3, 419–431 (1968).

    Google Scholar 

  29. A. V. Kuzhel' and D. V. Tret'yakov, “On a generalization of the Lax-Phillips scheme in scattering theory,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 2, 19–21 (1982).

    Google Scholar 

  30. A. V. Kuzhel' and D. V. Tret'yakov, “The generalized Lax-Phillips scheme in scattering theory,” in:Dynamical Systems (A Collection of Papers) [in Russian], Issue 2, (1983) pp. 115–121.

  31. L. A. Sakhnovich, “On the reduction of non-self-adjoint operators with continuous spectrum to diagonal form,”Mat. Sb.,44, No. 4, 509–548 (1957).

    Google Scholar 

  32. L. A. Sakhnovich, “On the reduction of a non-self-adjoint operator with continuous spectrum to diagonal form,”Usp. Mat. Nauk,13, No. 4, 193–196 (1958).

    Google Scholar 

  33. M. S. Brodskii, “The characteristic matrix functions of linear operators,”Mat. Sb.,39, No. 2, 179–200 (1956).

    Google Scholar 

  34. M. S. Brodskii,Triangular and Jordan Representations of Operators [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  35. A. G. Rutkas, “On characteristic functions of unbounded operators,”Teor. Funkts. Funkts. Anal. Pril., Issue 12, 20–35 (1970).

    Google Scholar 

  36. I. I. Karpenko, “Characteristic operator functions ofK Ioperators,”Dokl. Akad. Nauk Ukr, SSR. Ser. A, No. 9, 3–5 (1983).

    Google Scholar 

  37. A. V. Kuzhel', “On the reduction of unbounded non-self-adjoint operators to triangular form,”Dokl. Akad. Nauk SSSR,119, No. 5, 868–871 (1958).

    Google Scholar 

  38. A. V. Kuzhel', “Spectral analysis of unbounded operators which are not self-adjoint,”Dokl. Akad. Nauk SSSR,125, No. 1, 36–37 (1959).

    Google Scholar 

  39. A. V. Kuzhel',Spectral Analysis of Unbounded Operators Which Are Not Self-Adjoint [in Russian], Candidate Degree Thesis (Physics and Mathematics), Khar'kov (1959).

  40. A. V. Shtraus, “On the characteristic functions of linear operators,”Dokl. Akad. Nauk SSSR,126, No. 3, 514–516 (1959).

    Google Scholar 

  41. A. V. Shtraus, “Characteristic functions of linear operators,”Izv. Akad. Nauk SSSR. Ser. Mat.,24, No. 1, 43–47 (1960).

    Google Scholar 

  42. E. R. Tsekanovskii and Yu. L. Shmul'yan, “Theory of biextensions of operators in the rigged Hilbert spaces. Unbounded operator knots and characteristic functions,”Usp. Mat. Nauk,32, No. 5, 69–124 (1977).

    Google Scholar 

  43. Yu. M. Berezanskii, G. F. Us, and Z. G. Sheftel',Functional Analysis [in Russian], Vyshcha Shkola, Kiev (1990).

    Google Scholar 

  44. A. G. Rutkas, “To the theory of characteristic functions of linear operators,”Dokl. Akad. Nauk SSSR,229, No. 3, 546–549 (1976).

    Google Scholar 

  45. A. G. Rutkas, “The characteristic function and the model of a linear operator bundle,”Teor. Funkts. Funkts. Anal. Pril., Issue 45, 98–111 (1986).

    Google Scholar 

  46. A. G. Rutkas,Operator-Differential Equations, Unsolved with Respect to the Derivative, in Radiophysics [in Russian], Doctora Degree Thesis (Physics and Mathematics), Khar'kov (1988).

  47. Do Khong Tan, “On the multiplication theorem for characteristic functions of unbounded operators,”Teor. Funkts. Funkts. Anal. Pril., Issue 7, 65–74 (1969).

  48. A. V. Kuzhel', “Spectral analysis of unbounded non-self-adjoint operators in a space with indefinite metric,”Dokl. Akad. Nauk SSSR,178, No. 1, 31–34 (1968).

    Google Scholar 

  49. G. M. Gubreev, “Definition of the characteristic function of a W-knot and its fundamental properties,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 1, 3–6 (1978).

    Google Scholar 

  50. A. V. Shtraus, “To the theory of Hermitian operators,”Dokl. Akad. Nauk SSSR,67, No. 4, 211–214 (1949).

    Google Scholar 

  51. A. V. Shtraus, “To the theory of generalized resolvents of a symmetric operator,”Dokl. Akad. Nauk SSSR,78, No. 2, 217–220 (1951).

    Google Scholar 

  52. A. V. Shtraus, “On the self-adjoint operators in the orthogonal sum of Hilbert spaces,”Dokl. Akad. Nauk SSSR,144, No. 3, 512–515 (1962).

    Google Scholar 

  53. A. V. Shtraus, “On the extensions and characteristic function of a symmetric operator,”Izv. Akad. Nauk SSSR, Ser. Mat.,32, No. 1, 186–207 (1968).

    Google Scholar 

  54. A. V. Shtraus, “Extensions and generalized resolvents of a nondensely defined symmetric operator,”Izv. Akad. Nauk SSSR, Ser. Mat.,34, No. 1, 175–202 (1970).

    Google Scholar 

  55. A. V. Kuzhel' and L. I. Rudenko, “Tame extensions of Hermitian and isometric operators,”Ukr. Mat. Zh.,33, No. 6, 810–814 (1981).

    Google Scholar 

  56. A. V. Kuzhel' and L. I. Rudenko, “Description of tame extensions of Hermitian operators,”Funkts. Anal. Prilozh.,16, No. 1, 74–75 (1982).

    Google Scholar 

  57. A. V. Kuzhel',Extensions of Hermitian Operators [in Russian], Vyshcha Shkola, Kiev (1989).

    Google Scholar 

  58. A. N. Kochubei, “On characteristic functions of symmetric operators and their extensions,”Izv. Akad. Nauk Arm. SSR,15, No. 3, 219–232 (1980).

    Google Scholar 

  59. M. O. Talyush, “The typical structure of dissipative operators,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 11, 993–996 (1973).

    Google Scholar 

  60. A. N. Kochubei, “On the extensions of symmetric operators and symmetric binary relations,”Mat. Zametki,17, No. 1, 41–48 (1975).

    Google Scholar 

  61. A. N. Kochubei, “On the spectrum of self-adjoint extensions of a symmetric operator,”Mat. Zametki,19, No. 3, 429–434 (1976).

    Google Scholar 

  62. A. N. Kochubei, “On the extensions of a nondensely defined symmetric operator,”Sib. Mat. Zh.,18, No. 2, 314–320 (1977).

    Google Scholar 

  63. V. M. Bruk, “On a class of boundary-value problems with the spectral parameter in the boundary condition,”Mat. Sb.,100, No. 2, 210–216 (1976).

    Google Scholar 

  64. V. M. Bruk, “On the extensions of symmetric relations,”Mat. Zametki,22, No. 6, 825–834 (1977).

    Google Scholar 

  65. V. M. Bruk, “On the extensions of a symmetric operator depending on the parameter,” in:Functional Analysis (A Collection of Papers) [in Russian], Issue 10, Ul'yanovsk Pedagogical Institute, Ul'yanovsk (1978) pp. 32–40.

    Google Scholar 

  66. A. N. Kochubei, “On symmetric operators commuting with a family of unitary operators,”Funkts. Anal. Prilozh.,13, No. 4, 77–78 (1979).

    Google Scholar 

  67. V. A. Derkach and M. M. Malamud,The Weyl Function of an Hermitian Operator and Its Connection with the Characteristic Function [in Russian], Preprint No. 85.9 (104), Institute of Physics and Technology, Ukrainian Academy of Sciences, Donetsk (1985).

    Google Scholar 

  68. V. A. Derkach, M. M. Malamud, and E. R. Tsekanovskii, “Sectoral extensions of a positive operator and the characteristic function,”Ukr. Mat. Zh.,41, No. 2, 151–158 (1989).

    Google Scholar 

  69. S. A. Kuzhel',The Spaces of Limiting Values and the Characteristic Functions of Hermitian Operators [in Russian], Deposited at UkrNIINTI No. 1710-Uk.89, Kiev (1989).

  70. S. A. Kuzhel', “On the spaces of limiting values and tame extensions of Hermitian operators,”Ukr. Mat. Zh.,42, No. 6, 854–857 (1990).

    Google Scholar 

  71. M. M. Malamud, “An approach to the theory of extensions of a nondensely defined Hermitian operator,”Dokl. Akad Nauk Ukr. SSR, Ser. A, No. 3, 20–26 (1990).

    Google Scholar 

  72. R. S. Phillips, “Extensions of dual subspaces invariant with respect to the algebra,”Mathematics. A Collection of Translations,8, No. 6, 81–108 (1964).

    Google Scholar 

  73. A. V. Kuzhel' and Yu. P. Moskaleva, “Extensions of Hermitian nondensely defined operators with restrictions,” in: Abstracts of XVI All-Union School on the Theory of Operators [in Russian], Nizhnii Novgorod (1991), p. 121.

  74. A. V. Shtraus, “On a class of regular operator functions,”Dokl. Akad. Nauk SSSR,70, No. 4, 577–580 (1950).

    Google Scholar 

  75. A. V. Shtraus, “On generalized resolvents of a symmetric operator,”Dokl. Akad. Nauk SSSR,71, No. 2, 241–244 (1950).

    Google Scholar 

  76. T. Ya. Azizov and I. S. Iokhvidov,Fundamentals of the Theory of Linear Operators in Spaces with an Indefinite Metric [in Russian], Nauka, Moscow (1986).

    Google Scholar 

  77. A. V. Kuzhel', “Spectral analysis of quasiunitary operators of the first rank in a space with an indefinite metric,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 8, 1001–1003 (1961).

    Google Scholar 

  78. A. V. Kuzhel', “A triangular model of theK Ioperators in a space with an indefinite metric,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 5, 572–574 (1962).

    Google Scholar 

  79. A. V. Kuzhel', “The characteristic matrix functions of quasiunitary operators of arbitrary rank in a space with an indefinite metric,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 9, 1135–1138 (1962).

    Google Scholar 

  80. A. V. Kuzhel', “Spectral analysis of bounded non-self-adjoint operators in a space with an indefinite metric,”Dokl. Akad. Nauk SSSR,151, No. 4, 772–774 (1963).

    Google Scholar 

  81. A. V. Kuzhel', “A case of the existence of invariant subspaces for quasiunitary operators in a space with an indefinite metric,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 5, 583–585 (1966).

    Google Scholar 

  82. A. V. Kuzhel', “The spectral analysis of quasiunitary operators in a space with an indefinite metric,”Teor. Funkts. Funkts. Anal. Pril, Issue 4, 3–27 (1967).

    Google Scholar 

  83. A. V. Kuzhel',Some Problems in the Spectral Theory of Linear Operators in Spaces with Definite and an Indefinite Metric [in Russian], Doctor Degree Thesis (Physics and Mathematics), Kiev (1969).

  84. A. V. Kuzhel', “Tame extensions of Hermitian operators in a space with an indefinite metric,”Dokl. Akad. Nauk SSSR,265, No. 5, 1059–1061 (1982).

    Google Scholar 

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  1. Simferopol' University, Simferopol'

    A. V. Kuzhel'

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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 731–743, June, 1993.

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Kuzhel', A.V. Evolution of the concept of the characteristic function of a linear operator. Ukr Math J 45, 803–815 (1993). https://doi.org/10.1007/BF01061433

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