Journal of Mathematical Sciences

, Volume 73, Issue 2, pp 141–242 | Cite as

The extension theory of Hermitian operators and the moment problem

  • V. A. Derkach
  • M. M. Malamud
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Literature Cited

  1. 1.
    E. L. Aleksandrov, “On resolvents of symmetric nondensely defined operators,”Izv. Vuzov., Mat., No. 7, 3–12 (1970).MATHGoogle Scholar
  2. 2.
    E. L. Aleksandrov and G. M. Il’mushkin, “Generalized resolvents of symmetric operators,”Mat. Zamethi,19, No. 5, 783–794 (1976).Google Scholar
  3. 3.
    D. Z. Arov and L. Z. Grossman, “Scattering matrices in the theory of extensions of isometric operators,”Dokl. Akad. Nauk SSSR,270, No. 1, 17–20 (1983); English transl. in Soviet Math. Dokl., 27.MathSciNetGoogle Scholar
  4. 4.
    N. I. Akhiezer,The Classical Moment Problem [in Russian], Fizmatgiz, Moscow (1961).Google Scholar
  5. 5.
    N. I. Akhiezer and I. M. Glazman,Theory of Linear Operators in Hilbert Space [in Russian], Nauka, Moscow (1966).Google Scholar
  6. 6.
    Yu. M. Berezanskii,Expansions in Eigenfunctions of Self-Ajoint Operators, Amer. Math. Soc., Providence (1968).Google Scholar
  7. 7.
    M. Sh. Birman, “On self-adjoint extensions of positive definite operators,”Mat. Sb.,38, No. 4, 431–450 (1956).MATHMathSciNetGoogle Scholar
  8. 8.
    M. S. Brodskii,Triangular and Jordan Representations of Linear Operators, Amer. Math. Soc., Providence, Rhode Island (1971).Google Scholar
  9. 9.
    M. S. Brodskii and M. S. Livšic, “Spectral analysis of non-self-adjoint operators,”Uspekhi Mat. Nauk,13, No. 1, 3–85 (1958).Google Scholar
  10. 10.
    M. S. Chunaeva and A. N. Vernik, “The characteristic function of a linear relation in a space with an indefinite metric,”Funkts. Anal., No. 16, 42–52 (1981).MathSciNetGoogle Scholar
  11. 11.
    V. S. Vladimirov and B. I. Zav’yalov, “Automodel asymptotics of casual functions,”Teor. Mat. Fiz.,50, No. 2, 163–194 (1982).MathSciNetGoogle Scholar
  12. 12.
    V. I. Gorbachuk and M. L. Gorbachuk,Boundary Problems for Differential-Operator Equations [in Russian], Naukova Dumka, Kiev (1984).Google Scholar
  13. 13.
    V. I. Gorbachuk, M. L. Gorbachuk, and A. N. Kochubei, “Extension theory of symmetric operators and boundary-value problems for differential equations,”Ukr. Mat. Zh.,41, No. 10, 1299–1313 (1990).MathSciNetGoogle Scholar
  14. 14.
    V. A. Derkach, “Extensions of a Hermitian operator in a krein space,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 5, 5–9 (1988).MATHMathSciNetGoogle Scholar
  15. 15.
    V. A. Derkach, “On the extensions of a nondensely defined Hermitian operator in a Krein space,“Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 10, 14–18 (1990).Google Scholar
  16. 16.
    V. A. Derkach, “On generalized resolvents of a class of Hermitian operators in a Krein space,”Sov. Math. Dokl.,43, No. 2, 519–524 (1991).MATHMathSciNetGoogle Scholar
  17. 17.
    V. A. Derkach,Generalized Resolvents of Hermitian Operators in a Krein Space [in Russian], Preprint 92.2, IPMM Akad. Nauk Ukrainy (1992).Google Scholar
  18. 18.
    V. A. Derkach and M. M. Malamud,Weyl Function of Hermitian Operator and Its Connection with the Characteristic Function [in Russian], Preprint 85-9 (104), Fiz.-Tekhn. Inst. Akad. Nauk Ukrain. SSR. Donetsk (1985).Google Scholar
  19. 19.
    V. A. Derkach and M. M. Malamud, “On the Weyl function and Hermitian operators with gaps,”Sov. Math. Dokl.,35, No. 2, 393–398 (1987).MathSciNetGoogle Scholar
  20. 20.
    V. A. Derkach and M. M. Malamud,Generalized Resolvents and Boundary-Value Problems for Hermitian Operator with Gaps [in Russian], Preprint 88.59, Inst. Matem. Akad. Nauk USSR Kiev (1988).Google Scholar
  21. 21.
    V. A. Derkach and M. M. Malamud,On Some Classes of Solutions of the Moment Problem [in Russian], Manuscript No. 2239, Deposited at Ukr. Nauchn.-Issled. Inst. Nauchno-Tekhn. Informatsii, Kiev (1988).Google Scholar
  22. 22.
    V. A. Derkach and M. M. Malamud, “On some classes of analytic operator-valued functions with a non-negative imaginary part,”Dokl. Akad. Nauk. Ukr. SSR, Ser. A, No. 3, 13–17 (1989).MathSciNetGoogle Scholar
  23. 23.
    V. A. Derkach and M. M. Malamud, “The resolvent matrix of a Hermitian operator and a moment problem with gaps,”Sov. Math. Dokl.,42, No. 2, 429–435 (1991).MathSciNetGoogle Scholar
  24. 24.
    V. A. Derkach and M. M. Malamud, “The generalized resolvents of Hermitian operators and the truncated moment problem,”Dokl. Akad. Nauk Ukr., Ser. A, No. 11, 34–39 (1991).MathSciNetGoogle Scholar
  25. 25.
    V. A. Derkach and M. M. Malamud, “On a generalization of the Krein-Stieltjes class of functions,”Izv. Akad. Nauk Arm. SSR,26, No. 2, 115–137 (1991).MathSciNetGoogle Scholar
  26. 26.
    V. A. Derkach and M. M. Malamud, “Characteristic functions of almost solvable extensions of Hermitian operators,”Ukr. Mat. Zh.,44, No. 4, 435–459 (1992).MathSciNetGoogle Scholar
  27. 27.
    V. A. Derkach and M. M. Malamud, “Characteristic functions of linear operators,”Dokl. Rossiisk. Akad. Nauk,323, No. 5, 816–822 (1992).MathSciNetGoogle Scholar
  28. 28.
    V. A. Derkach and M. M. Malamud, “Inverse problems for Weyl functions, preresolvent and resolvent matrices of Hermitian operators,”Dokl. Rossiisk. Akad. Nauk,326, No. 1, 12–18 (1992).Google Scholar
  29. 29.
    T. Kato,Perturbation Theory for Linear Operators, Springer-Verlag (1966).Google Scholar
  30. 30.
    A. N. Kochubei, “On characteristic functions of symmetric operators and their extensions,”Sov. J. Contemporary Math. Anal., 15 (1980).Google Scholar
  31. 31.
    M. A. Krasnosel’skii, “On self-adjoint extensions of Hermitian operators,”Ukr. Mat. Zh.,1, 21–38 (1949).MathSciNetGoogle Scholar
  32. 32.
    M. G. Krein, “On Hermitian operator with defect index (1,1),”Dokl. Akad. Nauk SSSR,43, No. 8, 339–342 (1944).MathSciNetGoogle Scholar
  33. 33.
    M. G. Krein, “On the resolvents of a Hermitian operator with defect index (m, m),”Dokl. Akad. Nauk SSSR,52, No. 8, 657–660 (1946).MathSciNetGoogle Scholar
  34. 34.
    M. G. Krein, “The theory of self-adjoint extensions of semibounded Hermitian operators and its applications. I.,”Mat. Sb.,20, No. 3, 431–495 (1947).MATHMathSciNetGoogle Scholar
  35. 35.
    M. G. Krein, “On a generalization of Stieltjes investigations,”Dokl. Akad. Nauk SSSR,86, No. 6, 881–884 (1952).MathSciNetGoogle Scholar
  36. 36.
    M. G. Krein, “The description of solutions of the truncated moment problem,”Mat. Issledovaniya,2, No. 2, 114–132 (1967).MATHMathSciNetGoogle Scholar
  37. 37.
    M. G. Krein and G. K. Langer, “On defect subspaces and generalized resolvents of a Hermitian operator in the space IIx,”Funct. Anal. Appl.,5, 136–146, 217–228 (1971/1972).CrossRefGoogle Scholar
  38. 38.
    M. G. Krein and A. A. Nudelman,Markov Moment Problem and Extremal Problems, Amer. Math. Soc., Providence, Rhode Island (1977).Google Scholar
  39. 39.
    M. G. Krein and I. E. Ovcharenko, “On generalized resolvents and resolvent matrices of positive Hermitian operators,”Sov. Math. Dokl., 17 (1976).Google Scholar
  40. 40.
    M. G. Krein and I. E. Ovcharenko, “On theQ-functions andsc-resolvents of a nondensely defined Hermitian contraction,”Sib. Math. J., 18 (1977).Google Scholar
  41. 41.
    M. G. Krein and I. E. Ovcharenko, “Inverse problems forQ-functions and resolvent matrices of positive Hermitian operators,”Soviet. Math. Dokl., 19 (1978).Google Scholar
  42. 42.
    M. G. Krein and Sh. N. Saakyan, “Some new results in the theory of resolvents of Hermitian operators,”Soviet. Math. Dokl.,7, 1086–1089 (1966).Google Scholar
  43. 43.
    M. G. Krein and Sh. N. Saakyan, “The resolvent matrix of a Hermitian operator and characteristic functions related to it,”Funct. Anal. Appl., 4 (1970).Google Scholar
  44. 44.
    S. G. Krein,Linear Differential Equations in a Banach Space, Amer. Math. Soc., Providence, Rhode Island (1971).Google Scholar
  45. 45.
    S. G. Krein,Linear Equations in a Banach Space [in Russian], Nauka, Moscow (1971).Google Scholar
  46. 46.
    A. B. Kuzhel, “On a reduction of nonbounded non-self-adjoint operators to a triangular form,”Dokl. Akad. Nauk SSSR,119, No. 5, 868–871 (1958).MATHMathSciNetGoogle Scholar
  47. 47.
    P. Lankaster,Theory of Matrices, Academic Press, New York-London (1969).Google Scholar
  48. 48.
    M. S. Livšic, “On a spectral resolution of linear nonself-adjoint operators,” In:Amer. Math. Soc. Transl. (2), 5 (1957).Google Scholar
  49. 49.
    M. M. Malamud, “On extensions of Hermitian, sectorial operators, and dual pairs of contractions,”Sov. Math. Dokl.,39, No. 2 (1989).Google Scholar
  50. 50.
    M. M. Malamud, “Boundary-value problems for Hermitian operators with gaps,”Sov. Math. Dokl.,42, No. 1, 190–196 (1991).MathSciNetGoogle Scholar
  51. 51.
    M. M. Malamud, “On an approach to the extension theory of a nondensely defined Hermitian operator,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No.3, 20–25 (1990).MATHMathSciNetGoogle Scholar
  52. 52.
    M. M. Malamud, “On some classes of extensions of a Hermitian operator with gaps,”Ukr. Mat. Zh.,44, No. 2, 215–234 (1992).MATHMathSciNetGoogle Scholar
  53. 53.
    M. M. Malamud, “On the formula of generalized resolvents of a nondensely defined Hermitian operator,”Ukr. Mat. Zh.,44, No. 12, 1658–1688 (1992).MATHMathSciNetGoogle Scholar
  54. 54.
    M. A. Naimark, “Spectral functions of a symmetric operator,”Izv. Akad. Nauk SSSR, Ser. Mat.,4, No. 3, 277–318 (1940).MathSciNetGoogle Scholar
  55. 55.
    M. A. Naimark, “On spectral functions of a symmetric operator,”Izv. Akad. Nauk SSSR, Ser. Mat.,7, No. 6, 285–296 (1943).Google Scholar
  56. 56.
    B. S. Pavlov, “Extension theory and explicitly solvable models,”Uspekhi Mat. Nauk,42, No. 6, 99–131 (1987).MATHMathSciNetGoogle Scholar
  57. 57.
    F. S. Rofe-Beketov, “The numerical range of a linear relation and maximal relations,”Teor. Funkts. Funkts. Anal. Prilozhen.,44, 103–112 (1985).MATHGoogle Scholar
  58. 58.
    Sh. N. Saakyan “On the theory of resolvents of symmetric operators with infinite deficiency indices,”Dokl. Akad. Nauk Arm. SSR,41, 193–198 (1965).MATHGoogle Scholar
  59. 59.
    B. Sz.-Nagy, C. Foias,Harmonic Analysis of Operators in Hilbert Space, Paris and Akad. Kiado, Budapest (1967).Google Scholar
  60. 60.
    E. R. Tsekanovskii and Yu. L. Shmul’yan, “The theory of bi-extensions of operators on rigged Hilbert spaces. Unbounded operator and characteristic functions,”Russian Math. Surveys,32, 73–131 (1977).CrossRefGoogle Scholar
  61. 61.
    Yu. L. Shmul’yan, “The operator integral of Hellinger,”Amer. Math. Soc. Transl., (2),22 (1962).Google Scholar
  62. 62.
    Yu. L. Shmul’yan, “On a problem of generalized resolvents formula” [in Russian], Odessa Institute of Marine Engeneers, Odessa, (1969), pp. 269–271.Google Scholar
  63. 63.
    Yu. L. Shmul’yan, “Direct and inverse problems for resolvent matrices,”Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 6, 514–517 (1970).Google Scholar
  64. 64.
    Yu. L. Shmul’yan, “Regular and singular Hermitian operators,”Mat. Zametki,8, No. 2, 197–203 (1970).MathSciNetGoogle Scholar
  65. 65.
    A. V. Shtraus, “Generalized resolvents of symmetric operators,”Izv. Akad. Nauk SSSR, Ser. Mat.,18, No. 1, 51–86 (1954).MATHGoogle Scholar
  66. 66.
    A. V. Shtraus, “On multiplication theorem for characteristic functions of linear operators,”Dokl. Akad. Nauk SSSR,126, No. 4, 723–726 (1959).MathSciNetGoogle Scholar
  67. 67.
    A. V. Shtraus, “Characteristic functions of linear operators,”Amer. Math. Soc. Transl., (2),40, 1–37 (1964).MATHGoogle Scholar
  68. 68.
    A. V. Shtraus, “Extensions and generalized resolvents of nondensely defined symmetric operators,”Math. USSR Izv.,4, 179–208 (1970).MATHGoogle Scholar
  69. 69.
    A. V. Shtraus, “On the theory of extremal extensions of a bounded positive operator,”Funkts. Anal., No. 18, 115–126 (1982).MATHMathSciNetGoogle Scholar
  70. 70.
    A. V. Shtraus, “Generalized resolvents of bounded symmetric operators,”Funkts. Anal., No. 27, 187–196 (1987).MATHMathSciNetGoogle Scholar

Publications in other languages

  1. 71.
    D. Alpay, P. Bruinsma, A. Dijksma, and H. S. V. de Snoo, “A Hilbert space associated with a Nevanlinna function,” In:Proc. MTNS Meeting, Amsterdam (1989), pp. 115–122.Google Scholar
  2. 72.
    D. Alpay, P. Bruinsma, A. Dijksma and H. S. V. de Snoo, “Interpolation problems, extensions of symmetric operators, and reproducing kernel spaces,”Operator Theory: Advances and Applications,50, 35–82 (1991).Google Scholar
  3. 73.
    T. Ando and K. Nishio, “Positive self-adjoint extensions of positive symmetric operators,”Tohoku Math. J.,22, 65–75 (1970).MathSciNetGoogle Scholar
  4. 74.
    C. Bennewitz, “Symmetric relations on a Hilbert space,”Lect. Notes Math.,280, 212–218 (1972).MATHMathSciNetGoogle Scholar
  5. 75.
    L. de Branges,Hilbert Spaces of Entire Functions, Prentice Hall, Englewood Cliffs, New Jersey (1968).Google Scholar
  6. 76.
    R. C. Brown, “Notes on generalized boundary value problems in Banach spaces. I. Adjoint and Extension Theory,”Pacif. J. Math.,85, No. 2, 295–322 (1979).MATHGoogle Scholar
  7. 77.
    E. A. Coddington, “Extension theory of formally normal and symmetric subspaces,”Mem. Amer. Math. Soc.,134, 1–80 (1973).MathSciNetGoogle Scholar
  8. 78.
    E. A. Coddington and H. S. V. de Snoo, “Positive self-adjoint extensions of positive symmetric subspaces,”Math. Z.,159, 203–214 (1978).CrossRefMathSciNetGoogle Scholar
  9. 79.
    V. A. Derkach and M. M. Malamud, “Generalized resolvents and the boundary value problems for Hermitian operators with gaps,”J. Funct. Anal.,95, No. 1, 1–95 (1991).MathSciNetGoogle Scholar
  10. 80.
    A. Dijksma and H. S. V. de Snoo, “Self-adjoint extension of symmetric subspaces,”Pacif. J. Math.,54, No. 1, 71–100 (1974).Google Scholar
  11. 81.
    M. G. Krein and H. Langer, “Uber dieQ-Function eines π-hermiteschen Operators im Raume Πϰ,”Acta Sci. Math. Szeged,34, 191–230 (1973).MathSciNetGoogle Scholar
  12. 82.
    H. Langer, “Verallgemeinerte Resolventen eines-J-nichtnegativen Operators mit endlichen Defect,”J. Funct. Anal.,8, 287–320 (1971).CrossRefMATHGoogle Scholar
  13. 83.
    H. Langer and B. Textorius, “On generalized resolvents andQ-functions of symmetric linear relations (subspaces) in Hilbert space,”Pacif. J. Math.,72, No. 1, 135–165 (1977).MathSciNetGoogle Scholar
  14. 84.
    H. Langer and B. Textorius, “L-resolvent matrices of symmetric linear relations with equal defect numbers; applications to canonical differential relations,”Integral Equations Operator Theory,5, 208–243 (1982).CrossRefMathSciNetGoogle Scholar

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© Plenum Publishing Corporation 1995

Authors and Affiliations

  • V. A. Derkach
  • M. M. Malamud

There are no affiliations available

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