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The extension theory of Hermitian operators and the moment problem

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Literature Cited

  1. E. L. Aleksandrov, “On resolvents of symmetric nondensely defined operators,”Izv. Vuzov., Mat., No. 7, 3–12 (1970).

    MATH  Google Scholar 

  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. 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.

    MathSciNet  Google Scholar 

  4. N. I. Akhiezer,The Classical Moment Problem [in Russian], Fizmatgiz, Moscow (1961).

    Google Scholar 

  5. N. I. Akhiezer and I. M. Glazman,Theory of Linear Operators in Hilbert Space [in Russian], Nauka, Moscow (1966).

    Google Scholar 

  6. Yu. M. Berezanskii,Expansions in Eigenfunctions of Self-Ajoint Operators, Amer. Math. Soc., Providence (1968).

    Google Scholar 

  7. M. Sh. Birman, “On self-adjoint extensions of positive definite operators,”Mat. Sb.,38, No. 4, 431–450 (1956).

    MATH  MathSciNet  Google Scholar 

  8. M. S. Brodskii,Triangular and Jordan Representations of Linear Operators, Amer. Math. Soc., Providence, Rhode Island (1971).

    Google Scholar 

  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. 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).

    MathSciNet  Google Scholar 

  11. V. S. Vladimirov and B. I. Zav’yalov, “Automodel asymptotics of casual functions,”Teor. Mat. Fiz.,50, No. 2, 163–194 (1982).

    MathSciNet  Google Scholar 

  12. V. I. Gorbachuk and M. L. Gorbachuk,Boundary Problems for Differential-Operator Equations [in Russian], Naukova Dumka, Kiev (1984).

    Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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).

    MATH  MathSciNet  Google Scholar 

  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. 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).

    MATH  MathSciNet  Google Scholar 

  17. V. A. Derkach,Generalized Resolvents of Hermitian Operators in a Krein Space [in Russian], Preprint 92.2, IPMM Akad. Nauk Ukrainy (1992).

  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. 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).

    MathSciNet  Google Scholar 

  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. 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).

  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).

    MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  27. V. A. Derkach and M. M. Malamud, “Characteristic functions of linear operators,”Dokl. Rossiisk. Akad. Nauk,323, No. 5, 816–822 (1992).

    MathSciNet  Google Scholar 

  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. T. Kato,Perturbation Theory for Linear Operators, Springer-Verlag (1966).

  30. A. N. Kochubei, “On characteristic functions of symmetric operators and their extensions,”Sov. J. Contemporary Math. Anal., 15 (1980).

  31. M. A. Krasnosel’skii, “On self-adjoint extensions of Hermitian operators,”Ukr. Mat. Zh.,1, 21–38 (1949).

    MathSciNet  Google Scholar 

  32. M. G. Krein, “On Hermitian operator with defect index (1,1),”Dokl. Akad. Nauk SSSR,43, No. 8, 339–342 (1944).

    MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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).

    MATH  MathSciNet  Google Scholar 

  35. M. G. Krein, “On a generalization of Stieltjes investigations,”Dokl. Akad. Nauk SSSR,86, No. 6, 881–884 (1952).

    MathSciNet  Google Scholar 

  36. M. G. Krein, “The description of solutions of the truncated moment problem,”Mat. Issledovaniya,2, No. 2, 114–132 (1967).

    MATH  MathSciNet  Google Scholar 

  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).

    Article  Google Scholar 

  38. M. G. Krein and A. A. Nudelman,Markov Moment Problem and Extremal Problems, Amer. Math. Soc., Providence, Rhode Island (1977).

    Google Scholar 

  39. M. G. Krein and I. E. Ovcharenko, “On generalized resolvents and resolvent matrices of positive Hermitian operators,”Sov. Math. Dokl., 17 (1976).

  40. M. G. Krein and I. E. Ovcharenko, “On theQ-functions andsc-resolvents of a nondensely defined Hermitian contraction,”Sib. Math. J., 18 (1977).

  41. M. G. Krein and I. E. Ovcharenko, “Inverse problems forQ-functions and resolvent matrices of positive Hermitian operators,”Soviet. Math. Dokl., 19 (1978).

  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. 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).

  44. S. G. Krein,Linear Differential Equations in a Banach Space, Amer. Math. Soc., Providence, Rhode Island (1971).

    Google Scholar 

  45. S. G. Krein,Linear Equations in a Banach Space [in Russian], Nauka, Moscow (1971).

    Google Scholar 

  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).

    MATH  MathSciNet  Google Scholar 

  47. P. Lankaster,Theory of Matrices, Academic Press, New York-London (1969).

    Google Scholar 

  48. M. S. Livšic, “On a spectral resolution of linear nonself-adjoint operators,” In:Amer. Math. Soc. Transl. (2), 5 (1957).

  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. M. M. Malamud, “Boundary-value problems for Hermitian operators with gaps,”Sov. Math. Dokl.,42, No. 1, 190–196 (1991).

    MathSciNet  Google Scholar 

  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).

    MATH  MathSciNet  Google Scholar 

  52. M. M. Malamud, “On some classes of extensions of a Hermitian operator with gaps,”Ukr. Mat. Zh.,44, No. 2, 215–234 (1992).

    MATH  MathSciNet  Google Scholar 

  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).

    MATH  MathSciNet  Google Scholar 

  54. M. A. Naimark, “Spectral functions of a symmetric operator,”Izv. Akad. Nauk SSSR, Ser. Mat.,4, No. 3, 277–318 (1940).

    MathSciNet  Google Scholar 

  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. B. S. Pavlov, “Extension theory and explicitly solvable models,”Uspekhi Mat. Nauk,42, No. 6, 99–131 (1987).

    MATH  MathSciNet  Google Scholar 

  57. F. S. Rofe-Beketov, “The numerical range of a linear relation and maximal relations,”Teor. Funkts. Funkts. Anal. Prilozhen.,44, 103–112 (1985).

    MATH  Google Scholar 

  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).

    MATH  Google Scholar 

  59. B. Sz.-Nagy, C. Foias,Harmonic Analysis of Operators in Hilbert Space, Paris and Akad. Kiado, Budapest (1967).

    Google Scholar 

  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).

    Article  Google Scholar 

  61. Yu. L. Shmul’yan, “The operator integral of Hellinger,”Amer. Math. Soc. Transl., (2),22 (1962).

  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. 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. Yu. L. Shmul’yan, “Regular and singular Hermitian operators,”Mat. Zametki,8, No. 2, 197–203 (1970).

    MathSciNet  Google Scholar 

  65. A. V. Shtraus, “Generalized resolvents of symmetric operators,”Izv. Akad. Nauk SSSR, Ser. Mat.,18, No. 1, 51–86 (1954).

    MATH  Google Scholar 

  66. A. V. Shtraus, “On multiplication theorem for characteristic functions of linear operators,”Dokl. Akad. Nauk SSSR,126, No. 4, 723–726 (1959).

    MathSciNet  Google Scholar 

  67. A. V. Shtraus, “Characteristic functions of linear operators,”Amer. Math. Soc. Transl., (2),40, 1–37 (1964).

    MATH  Google Scholar 

  68. A. V. Shtraus, “Extensions and generalized resolvents of nondensely defined symmetric operators,”Math. USSR Izv.,4, 179–208 (1970).

    MATH  Google Scholar 

  69. A. V. Shtraus, “On the theory of extremal extensions of a bounded positive operator,”Funkts. Anal., No. 18, 115–126 (1982).

    MATH  MathSciNet  Google Scholar 

  70. A. V. Shtraus, “Generalized resolvents of bounded symmetric operators,”Funkts. Anal., No. 27, 187–196 (1987).

    MATH  MathSciNet  Google Scholar 

Publications in other languages

  1. 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.

  2. 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. T. Ando and K. Nishio, “Positive self-adjoint extensions of positive symmetric operators,”Tohoku Math. J.,22, 65–75 (1970).

    MathSciNet  Google Scholar 

  4. C. Bennewitz, “Symmetric relations on a Hilbert space,”Lect. Notes Math.,280, 212–218 (1972).

    MATH  MathSciNet  Google Scholar 

  5. L. de Branges,Hilbert Spaces of Entire Functions, Prentice Hall, Englewood Cliffs, New Jersey (1968).

    Google Scholar 

  6. 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).

    MATH  Google Scholar 

  7. E. A. Coddington, “Extension theory of formally normal and symmetric subspaces,”Mem. Amer. Math. Soc.,134, 1–80 (1973).

    MathSciNet  Google Scholar 

  8. E. A. Coddington and H. S. V. de Snoo, “Positive self-adjoint extensions of positive symmetric subspaces,”Math. Z.,159, 203–214 (1978).

    Article  MathSciNet  Google Scholar 

  9. 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).

    MathSciNet  Google Scholar 

  10. 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. M. G. Krein and H. Langer, “Uber dieQ-Function eines π-hermiteschen Operators im Raume Π ϰ ,”Acta Sci. Math. Szeged,34, 191–230 (1973).

    MathSciNet  Google Scholar 

  12. H. Langer, “Verallgemeinerte Resolventen eines-J-nichtnegativen Operators mit endlichen Defect,”J. Funct. Anal.,8, 287–320 (1971).

    Article  MATH  Google Scholar 

  13. 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).

    MathSciNet  Google Scholar 

  14. 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).

    Article  MathSciNet  Google Scholar 

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 5, Analiz-3, 1993.

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Derkach, V.A., Malamud, M.M. The extension theory of Hermitian operators and the moment problem. J Math Sci 73, 141–242 (1995). https://doi.org/10.1007/BF02367240

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