Skip to main content
SpringerLink
Log in

Boundary Triplets, Weyl Functions, and the Kreĭn Formula

  • Reference work entry
  • First Online:
Operator Theory

Abstract

This chapter contains a short review of the theory of boundary triplets, and the corresponding Weyl functions, of symmetric operators in Hilbert and Kreĭn spaces. The theory of generalized resolvents of such operators is exposed from the point of view of boundary triplets approach. Applications to different continuation problems related to the extension theory of Kreĭn space symmetric operators are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 999.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 549.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Achieser, N.I.: The Classical Moment Problem. Fizmatgiz, Moscow (1961)

    Google Scholar 

  2. Achieser, N.I., Glasmann, I.M.: Theorie der Linearen Operatoren im Hilbertraum, 8th edn. Akademie, Berlin (1981)

    Google Scholar 

  3. Adamyan, V.M., Arov, D.Z., Kreĭn, M.G.: Analytic properties of the Schmidt pairs of a Hankel operator and the generalized Schur-Takagi problem. Mat. Sb. 86, 34–75 (1971)

    MathSciNet  Google Scholar 

  4. Alpay, D., Dym, H.: On applications of reproducing kernel spaces to the Schur algorithm and rational J unitary factorization. In: Gohberg, I. (ed.) I. Schur Methods in Operator and Signal Processing. Operator Theory: Advances and Applications, vol. 18, pp. 89–159. Birkhäuser, Basel (1986)

    Chapter  Google Scholar 

  5. Alpay, D., Bolotnikov, V., Dijksma, A.: On the Nevanlinna-Pick interpolation problem for generalized Stieltjes functions. Integr. Equ. Oper. Theory 30, 379–408 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alpay, D., Dijksma, A., Langer, H.: The transformation of Issai Schur and related topics in an indefinite setting. In: Operator Theory: Advances and Applications, vol. 176, pp. 1–98. Birkhäuser, Basel (2007)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Arlinski, Yu.: Boundary triplets and maximal accretive extensions of sectorial operators. In: Hassi, S., de Snoo, H.S.V., Szafraniec, F.H. (eds.) Operator Methods for Boundary Value Problems. London Mathematical Society Lecture Note Series, vol. 404, pp. 35–72. Cambridge University Press, Cambridge (2012)

    Chapter  Google Scholar 

  9. Arlinski, Yu., Tsekanovskii, E.: M. Krein’s research on semi-bounded operators, its contemporary developments, and applications. In: Adamyan, V., et al. (eds.) Modern Analysis and Applications. The Mark Krein Centenary Conference. Vol. 1: Operator Theory and Related Topics. Operator Theory: Advances and Applications, vol. 190, pp. 65–112. Birkhauser, Basel (2009)

    Google Scholar 

  10. Arlinski, Yu., Belyi, S., Tsekanovski, E.: Conservative Realizations of Herglotz-Nevanlinna Functions, 528 pp. Birkhäuser, Basel (2011)

    Book  Google Scholar 

  11. Arov, D.Z.: The generalized bitangent Caratheodory-Nevanlinna-Pick problem and (j, J 0)-inner matrix functions. Russ. Acad. Sci. Izvestiya Math. 42(1), 3–32 (1994)

    MathSciNet  Google Scholar 

  12. Arov, D.Z., Dym, H.: J-Contractive Matrix Valued Functions and Related Topics, 575 p. Cambridge University Press, Cambridge (2008)

    Book  Google Scholar 

  13. Arov, D.Z., Grossman, L.Z.: Scattering matrices in the theory of unitary extensions of isometric operators. Math. Nachr. 157, 105–123 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Azizov, T.Y., Iokhvidov, I.S.: Linear Operators in Spaces with Indefinite Metric. Wiley, New York (1989)

    Google Scholar 

  15. Baidiuk, D.: Boundary triplets and generalized resolvents of isometric operators in Pontryagin spaces. J. Math. Sci. 194(5), 513–531 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ball, J.A., Helton, J.W.: A Beurling–Lax theorem for the Lie group U(m, n) which contains most classical interpolation theory. J. Oper. Theory 9, 107–142 (1983)

    MathSciNet  MATH  Google Scholar 

  17. Ball, J.A., Gohberg, I.T., Rodman, L.: Interpolation of Rational Matrix Functions, OT45. Birkhäuser, Basel (1990)

    Book  Google Scholar 

  18. Behrndt, J., Langer, M.: Boundary value problems for elliptic partial differential operators on bounded domains. J. Funct. Anal. 243, 536–565 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Behrndt, J., Luger, A., Trunk, C.: Generalized resolvents of a class of symmetric operators in Krein spaces. Oper. Theory Adv. Appl. 175, 13–32 (2007)

    Article  MathSciNet  Google Scholar 

  20. Behrndt, J., Derkach, V., Hassi, S., de Snoo, H.S.V.: A realization theorem for generalized Nevanlinna families. Oper. Matrices 5(4), 679–706 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Benewitz, C.: Symmetric relations on a Hilbert space. Lect. Notes Math. 280, 212–218 (1972)

    Article  Google Scholar 

  22. Birman, M.Sh.: On the self-adjoint extensions of positive definite operators. Math. Sb. 38, 431–450 (1956)

    MathSciNet  Google Scholar 

  23. Bognar, J.: Indefinite inner product spaces. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 78, ix+224 pp. Springer, New York (1974)

    Google Scholar 

  24. Brodskii, M.S.: Triangular and Jordan Representations of Linear Operators, 287 pp. Nauka, Moscow (1969)

    Google Scholar 

  25. Calkin, J.W.: Abstract symmetric boundary conditions. TAMS 45(3), 369–442 (1939)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Derevyagin, M.: On the Schur algorithm for indefinite moment problem. Meth. Funct. Anal. Topol. 9(2), 133–145 (2003)

    MathSciNet  MATH  Google Scholar 

  28. Derevyagin, M., Derkach, V.: Spectral problems for generalized Jacobi matrices. Linear Algebra Appl. 382, 1–24 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Derkach, V.: On Weyl function and generalized resolvents of a Hermitian operator in a Kreĭn space. Integr. Equ. Oper. Theory 23, 387–415 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Derkach, V.: On indefinite moment problem and resolvent matrices of Hermitian operators in Kreĭn spaces. Math. Nachr. 184, 135–166 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Derkach, V.: On Kreĭn space symmetric linear relations with gaps. Meth. Funct. Anal. Topol. 4(2), 16–40 (1998)

    MathSciNet  MATH  Google Scholar 

  32. Derkach, V.: On generalized resolvents of Hermitian relations in Krein spaces. Funct. Anal. J. Math. Sci. (NY) 97(5), 4420–4460 (1999)

    Google Scholar 

  33. Derkach, V., Malamud, M.: On Weyl function and Hermitian operators with gaps. Doklady Akad. Nauk. SSSR 293(5), 1041–1046 (1987)

    MathSciNet  Google Scholar 

  34. Derkach, V., Malamud, M.: Generalized resolvents and the boundary value problems for hermitian operators with gaps. J. Funct. Anal. 95, 1–95 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  35. Derkach, V., Malamud, M.: The extension theory of Hermitian operators and the moment problem. J. Math. Sci. 73, 141–242 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. Derkach, V., Hassi, S., Malamud, M., de Snoo, H.: Generalized resolvents of symmetric operators and admissibility. Meth. Funct. Anal. Topol. 6(3), 24–55 (2000)

    MATH  Google Scholar 

  37. Derkach, V., Hassi, S., Malamud, M., de Snoo, H.: Boundary relations and Weyl families. Trans. Am. Math. Soc. 358, 5351–5400 (2006)

    Article  MATH  Google Scholar 

  38. Derkach, V., Hassi, S., Malamud, M., de Snoo, H.: Boundary relations and generalized resolvents of symmetric operators. Russ. J. Math. Phys. 16(1), 17–60 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Derkach, V., Hassi, S., Malamud, M., de Snoo, H.: Boundary triplets and Weyl functions. In: Hassi, S., de Snoo, H., Szafraniec, F.H. (eds.) Recent Developments, Operator Methods for Boundary Value Problems. LMS, Cambridge (2012)

    Google Scholar 

  40. Dijksma, A., Langer, H., de Snoo, H.: Symmetric Sturm-Liouville operator with eigenvalue depending boundary conditions. Can. Math. Soc. Conf. Proc. 8, 87–116 (1987)

    MathSciNet  Google Scholar 

  41. Dijksma, A., Langer, H., de Snoo, H.: Generalized coresolvents of standard isometric relations and generalized resolvents of standard symmetric relations in Kreĭn spaces. In: Operator Theory: Advances and Applications, vol. 48, pp. 261–274. Birkhäuser, Basel (1990)

    Google Scholar 

  42. Dym, H.: On Hermitian block Hankel matrices, matrix polynomials, the Hamburger moment problem, interpolation and maximum entropy. Int. Equ. Oper. Theory 12, 757–811 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  43. Dym, H.: J–Contractive Matrix Functions, Reproducing Kernel Hilbert Spaces and Interpolation. CBMS Regional Series in Mathematics, vol. 71. AMS, Providence (1989)

    Google Scholar 

  44. Fedchina, I.P.: A solvability criterion for tangential Nevanlinna-Pick problem. Mat. Issled. 26, 213–226 (1972)

    Google Scholar 

  45. Friedrichs, K.: Spectraltheorie halbbeschränkter operatoren. Math. Ann. 109, 405–487 (1934)

    Google Scholar 

  46. Gorbachuk, M.L.: Self-adjoint boundary value problems for differential equation of the second order with unbounded operator coefficient. Funct. Anal. Appl. 5(1), 10–21 (1971)

    Article  Google Scholar 

  47. Gorbachuk, M.L., Gorbachuk, V.I.: Boundary Problems for Differential Operator Equations. Naukova Dumka, Kiev (1984)

    MATH  Google Scholar 

  48. Grubb, G.: A characterization of the non local boundary value problems associated with an elliptic operator. Ann. Scuola Norm. Super Pisa 22(3), 425–513 (1968)

    MathSciNet  MATH  Google Scholar 

  49. Jonas, P.: A class of operator-valued meromorphic functions on the unit disc. Ann. Acad. Sci. Fenn. Ser. A. I. Math. 17, 257–284 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  50. Kac, I.S., Kreĭn, M.G.: R-functions – analytic functions mapping the upper halfplane into itself. Supplement to the Russian edition of F.V. Atkinson. In: Discrete and Continuous Boundary Problems. American Mathematical Society Translation Series, vol. 2, p. 103. Mir, Moscow (1974)

    Google Scholar 

  51. Kaltenbäck, M., Woracek, H.: The Krein formula for generalized resolvents in degenerated inner product spaces. Monatsh. Math. 127(2), 119–140 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  52. Katsnelson, V.E., Kheifets, A.Ya., Yuditskii, P.M.: The abstract interpolation problem and extension theory of isometric operators. In: Marchenko, V.A. (ed.) Operators in Spaces of Functions and Problems in Function Theory, pp. 83–96. Naukova Dumka, Kiev (1987)

    Google Scholar 

  53. Kochubei, A.N.: On extentions of symmetric operators and symmetric binary relations. Mat. Z. 17(1), 41–48 (1975)

    Google Scholar 

  54. Kreĭn, M.G.: Sur le probleme du prolongement des fonctions hermitiennes positives et continues. Dokl. Akad. Nauk. SSSR 26, 17–22 (1940)

    Google Scholar 

  55. Kreĭn, M.G.: On Hermitian operators with deficiency index (1, 1). Dokl. Akad. Nauk. SSSR 43(8), 339–342 (1944)

    Google Scholar 

  56. Kreĭn, M.G.: Fundamental aspects of the representation theory of Hermitian operators with deficiency index (m, m). Ukrain. Math. Zh. 1, 3–66 (1944) (Russian) [English translation: Am. Math. Soc. Transl. (2) 97, 75–143 (1970)]

    Google Scholar 

  57. Kreĭn, M.G.: On resolvents of Hermitian operator with deficiency index (m, m). Dokl. Akad. Nauk. SSSR 52, 657–660 (1946)

    Google Scholar 

  58. Kreĭn, M.G.: Theory of self-adjoint extensions of semibounded hermitian operators and applications, II. Mat. Sb. 21(3), 365–404 (1947)

    Google Scholar 

  59. Kreĭn, M.G.: On a generalization of Stieltjes investigations. Dokl. Akad. Nauk. SSSR 86(6), 881–884 (1952)

    Google Scholar 

  60. Kreĭn, M.G., Langer, H.: On defect subspaces and generalized resolvents of Hermitian operator in Pontryagin space. Funkts. Anal. Prilozhen 5(2), 59–71; ibid 5(3), 54–69 (1952) (Russian) [English translation: Funct. Anal. Appl. 5, 136–146 (1971); ibid 5, 217–228 (1971)]

    Google Scholar 

  61. Kreĭn, M.G., Langer, H.: Über die Q-function eines π-hermiteschen operators in Raume \(\Pi _{\kappa }\). Acta. Sci. Math. (Szeged) 34, 191–230 (1973)

    MathSciNet  MATH  Google Scholar 

  62. Kreĭn, M.G., Langer, H.: Über einige Fortsetzungsprobleme, die ung mit der Theorie hermitescher operatoren im Raume \(\Pi _{\kappa }\) zusammenhängen. I. Math. Nachr. 77, 187–236 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  63. Kreĭn, M.G., Langer, H.: On some extension problem which are closely connected with the theory of hermitian operators in a space \(\Pi _{\kappa }\) III. Indefinite analogues of the Hamburger and Stieltjes moment problems, Part I. Beiträge Zur Anal. 14, 25–40 (1979)

    Google Scholar 

  64. Kreĭn, M.G., Nudelman, A.A.: Markov Moment Problem and Extremal Problems. American Mathematical Society, Providence (1977)

    MATH  Google Scholar 

  65. Kreĭn, M.G., Ovčarenko, I.E.: On generalized resolvents and resolvent matrices of positive Hermitian operators. Soviet Math. Dokl. 231, 1063–1066 (1976)

    Google Scholar 

  66. Kreĭn, M.G., Saakyan, Sh.N.: Resolvent matrix of a Hermitian operator and charachteristic functions. Funct. Anal. Appl. 4(3), 103–104 (1970)

    Google Scholar 

  67. Lance, W.E., Storozh, O.G.: Methods of Theory of Unbounded Operators (in Russian), p. 212. Nauk Dumka, Kiev (1983)

    Google Scholar 

  68. Langer, H.: Verallgemeinerte resolventen eines J–nichtnegativen operators mit endlichen defect. J. Funct. Anal. 8, 287–320 (1971)

    Article  MATH  Google Scholar 

  69. Langer, H.: Spectral functions of definitizable operators in Kreĭn spaces. In: Functional Analysis, Proceedings of a Conference held at Dubrovnik. Lecture Notes in Mathematics, vol. 948. Springer, Berlin (1982)

    Google Scholar 

  70. Langer, H., Sorjonen, P.: Verallgemeinerte resolventen hermitescher und isometrischer operatoren im pontrjaginraum. Ann. Acad. Sci. Fenn. Ser. A 561, 3–45 (1974)

    MathSciNet  Google Scholar 

  71. Langer, H., Textorius, B.: On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space. Pacific J. Math. 72, 135–165 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  72. Livsič, M.S.: On a certain class of linear operators in Hilbert space. Rec. Math. [Mat. Sb.] N.S. 19(61/2), 239–262 (1946)

    Google Scholar 

  73. Malamud, M.M.: On the formula of generalized resolvents of a nondensely defined Hermitian operator. Ukrain. Math. Zh. 44(2), 1658–1688 (1992)

    MATH  Google Scholar 

  74. Malamud, M.M.: Spectral theory of elliptic operators in exterior domains. Russ. J. Math. Phys. 17(1), 96–125 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  75. Malamud, M.M., Mogilevskii, V.I.: Krein type formula for canonical resolvents of dual pairs of linear relations. Meth. Funct. Anal. Topol. 8(4), 72–100 (2002)

    MathSciNet  MATH  Google Scholar 

  76. Mogilevskii, V.I.: Boundary triplets and Kreĭn type resolvent formula for symmetric operators with unequal defect numbers. Meth. Funct. Anal. Topol. 12(3), 258–280 (2006)

    MathSciNet  MATH  Google Scholar 

  77. Naĭmark, M.A.: On spectral functions of a symmetric operator. Izv. Akad. Nauk. SSSR Ser. Mat. 7, 285–296 (1943)

    MATH  Google Scholar 

  78. Nevanlinna, R.: Uber beschränkte Funktionen, die in gegebene Punkten forgeschriebene Werte annehmen. Ann. Acad. Sci. Fenn. 13(1), 1–71 (1919)

    MathSciNet  Google Scholar 

  79. Nudelman, A.A.: A new problem of the type of the moment problem (Russian). Dokl. Akad. Nauk. SSSR 233(5), 792–795 (1977)

    MathSciNet  Google Scholar 

  80. Pick, G.: Uber die beschrankungen analytischer Funktionen welche durch vorgegebene Funktionswerte hewirkt sind. Math. Ann. 77, 7–23 (1916)

    Article  Google Scholar 

  81. Rofe-Beketov, F.S.: On self-adjoint extensions of differential operators in a space of vector-functions. Teor. Funkts. Funkts. Anal. Prilozhen 8, 3–24 (1969)

    MathSciNet  MATH  Google Scholar 

  82. Saakyan, Sh.N.: Theory of resolvents of a symmetric operator with infinite defect numbers (Russian). Akad. Nauk Armjan. SSSR Dokl. 41, 193–198 (1965)

    MATH  Google Scholar 

  83. Shmul’jan, Yu.L.: Extension theory for operators and spaces with indefinite metric. Izv. Akad. Nauk. SSSR Ser. Mat. 38(4), 896–908 (1974)

    MathSciNet  Google Scholar 

  84. Štrauss, A.V.: Generalized resolvents of symmetric operators. Izv. Akad. Nauk. SSSR Ser. Mat. 18, 51–86 (1954) (Russian) (English translation: Math. USSR-Izvestija 4, 179–208)

    Google Scholar 

  85. Štrauss, A.V.: Characteristic functions of linear operators. Izv. Akad. Nauk. SSSR Ser. Mat. 24(1), 24–34 (1960)

    Google Scholar 

  86. Štrauss, A.V.: Extensions and generalized resolvents of a symmetric operator which is not densely defined. Izv. Akad. Nauk. SSSR Ser. Mat. 34, 175–202 (1970) (Russian) (English translation: Math. USSR-Izvestija 4, 179–208)

    Google Scholar 

  87. Sz.-Nagy, B., Koranyi, A.: Relations d’un probleme de Nevanlinna et Pick avee la theorie des operations de l’espace hilbertien. Acta Math. Acad. Sci. Hung. 7, 295–303 (1956)

    Google Scholar 

  88. Višik, M.I.: On general boundary problems for elliptic differential equations (Russian). Trudy Moskov. Mat. Obšc. 1, 187–246 (1952)

    Google Scholar 

  89. von Neumann, J.: Über adjungierte operatoren. Ann. Math. 33(2), 294–310 (1932)

    Article  Google Scholar 

  90. Woracek, H.: Nevanlinna-Pick interpolation: the degenerated case. Linear Algebra Appl. 252, 141–158 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Donetsk National University, Donetsk, Ukraine

    Vladimir Derkach

Authors
  1. Vladimir Derkach
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimir Derkach .

Editor information

Editors and Affiliations

  1. Earl Katz Chair in Algebraic System Theory, Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Daniel Alpay

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer Basel

About this entry

Cite this entry

Derkach, V. (2015). Boundary Triplets, Weyl Functions, and the Kreĭn Formula. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0667-1_32

Download citation

Publish with us

Policies and ethics

Search

Navigation