Skip to main content
SpringerLink
Log in

Unitary colligations in Krein spaces and their role in the extension theory of isometries and symmetric linear relations in Hilbert spaces

  • Chapter
  • First Online:
Functional Analysis II

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1242))

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 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.00
Price excludes VAT (USA)
  • Compact, lightweight 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D.Z. Arov, Passive linear stationary dynamical systems, Sibirskii Matem. Ž. 20:2(114) (1979), 211–228.

    MathSciNet  MATH  Google Scholar 

  2. T. Ya Azizov, On the theory of extensions of isometric and symmetric operators in spaces with an indefinite metric, Preprint Voronesh University, 1982; Deposited paper no. 3420–82.

    Google Scholar 

  3. T. Ya Azizov, Extensions of J-isometric and J-unitary operators, Functional Anal. i Pr. ložen 18 (1984), 57–58 (English translation: Functional Anal. Appl. (1984), 46–48).

    MathSciNet  MATH  Google Scholar 

  4. J. Bognar, Indefinite Inner Product Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1974.

    Book  MATH  Google Scholar 

  5. M.S. Brodskii, Unitary operator colligations and their characteristic functions, Uspekhi Mat. Nauk 33:4 (1978), 141–168 (English translation: Russian Math. Surveys 33:4 (1978), 159–191).

    MathSciNet  Google Scholar 

  6. M.S. Cunaeva and A.N. Vernik, The characteristic function of a linear relation in a space with an indefinite metric, Functional Analysis, Ul'yanovsk. Gos. Ped. Inst., Ul'yanovsk, 16 (1981) 42–52.

    Google Scholar 

  7. C. Davis, J-unitary dilation of a general operator, Acta Sci. Math. (Szeged) 31 (1970), 75–86.

    MathSciNet  MATH  Google Scholar 

  8. J. Dieudonné, Quasi-hermitian operators. Proc. Internat. Symposium Linear Spaces, Jerusalem 1961.

    Google Scholar 

  9. A. Dijksma, H. Langer and H.S.V. de Snoo, Selfadjoint πk-extensions of symmetric subspaces: an abstract approach to boundary problems with spectral parameter in the boundary conditions, Integral equations and operator theory 7 (1984), 459–515.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Dijksma, H. Langer and H.S.V. de Snoo, Unitary colligations in πk-spaces, characteristic functions and Štraus extensions, to appear in Pacific J. Math.

    Google Scholar 

  11. A. Dijksma, H. Langer and H.S.V. de Snoo, Characteristic functions of unitary operator colligations in πk-spaces, to be published.

    Google Scholar 

  12. A. Dijksma, H. Langer and H.S.V. de Snoo, Representations of holomorphic functions by means of resolvents of unitary or selfadjoint operators in Krein spaces, to be published.

    Google Scholar 

  13. A. Dijksma and H.S.V. de Snoo, Selfadjoint extensions of symmetric subspaces, Pacific J. Math. 54 (1974), 71–100.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. Dijksma and H.S.V. de Snoo, Symmetric and selfadjoint relations in Krein spaces I, to be published.

    Google Scholar 

  15. I.C. Gohberg and M.G. Krein, Theory of Volterra operators in Hilbert space and its applications (Russian), Moscow 1967; English transl.: Amer. Math. Soc. Transl. of Math. Monographs, 24, 1970.

    Google Scholar 

  16. M.G. Krein and H. Langer, Uber die verallgemeinerten Resolventen und die charakteristische Funktion eines isometrischen Operators im Raume πk, Hilbert Space Operators and Operator Algebras (Proc. Int. Conf., Tihany, 1970) Colloqu. Math. Soc. János Bolyai, no. 5, North-Holland, Amsterdam (1972), 353–399.

    Google Scholar 

  17. M.G. Krein and H. Langer, Some propositions on analytic matrix functions related to the theory of operators in the space πk, Acta Sci. Math. (Szeged) 43 (1981), 181–205.

    MathSciNet  MATH  Google Scholar 

  18. H. Langer, Spectral functions of definitizable operators in Krein spaces in "Functional Analysis, Proceedings Dubrovnik 1981", Lecture Notes in Mathematics 948, Springer Verlag, Berlin-Heidelberg-New York, 1982.

    Google Scholar 

  19. H. Langer and P. Sorjonen, Verallgemeinerte Resolventen hermitescher und isometrischer Operatoren im Pontryaginraum, Ann. Acad. Sci. Fennicae A.I. Math. 561 (1974), 1–45.

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. H. Langer and B. Textorius, L-resolvent matrices of symmetric linear relations with equal defect numbers; applications to canonical differential relations, Integral equations and operator theory 5 (1982), 208–243.

    Article  MathSciNet  MATH  Google Scholar 

  22. H. Langer and B. Textorius, Generalized resolvents of contractions, Acta. Sci. Math. (Szeged) 44 (1982), 125–131.

    MathSciNet  MATH  Google Scholar 

  23. H. Langer and B. Textorius, Generalized resolvents of dual pairs of contractions, Invariant subspaces and other topics (Proc. 6-th Int. Conf. on Operator Theory, Romania, 1981) Birkhauser Verlag, Basel (1982), 103–118.

    Book  MATH  Google Scholar 

  24. P.D. Lax, Symmetrizable linear transformations, Comm. Pure Appl. Math. VII (1954), 633–647.

    Article  MathSciNet  MATH  Google Scholar 

  25. B. Orcutt, Canonical differential equations, University of Virginia Ph. D. Thesis, 1969.

    Google Scholar 

  26. V.P. Potapov, The multiplicative structure of J-contractive matrix functions, Trudy Moskov. Mat. Obšč. 4 (1955), 125–236 (Russian).

    MathSciNet  Google Scholar 

  27. W.T. Reid, Symmetrizable completely continuous linear transformations in Hilbert space, Duke Math. J. 18 (1951), 41–56.

    Article  MathSciNet  MATH  Google Scholar 

  28. A.V. Štraus, On some questions in the theory of symmetric operators, Moscow State University, Dissertation, 1960.

    Google Scholar 

  29. A.V. Štraus, Characteristic functions of linear operators, Izv. Akad. Nauk SSSR Ser. Mat. 24 (1960) 43–74 (English translation: Amer. Math. Soc. Transl. (2) 40 (1964) 1–37).

    MathSciNet  Google Scholar 

  30. A.V. Štraus, On the extensions of symmetric operators depending on a parameter, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965) 1389–1416 (English translation: Amer. Math. Soc. Transl. (2) 61 (1967) 113–141).

    MathSciNet  Google Scholar 

  31. A.V. Štraus, On one-parameter families of extensions of a symmetric operator, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966) 1325–1352 (English translation: Amer. Math. Soc. Transl. (2) 90 (1970) 135–164).

    MathSciNet  Google Scholar 

  32. A.V. Štraus, On the extensions and the characteristic function of a symmetric operator, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968) 186–207 (English translation: Math. USSR Izv. 2 (1968) 181–204).

    MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  34. B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert space, North-Holland Publishing Company, Amsterdam-London, 1970.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisch Instituut Rijksuniversiteit Groningen, Postbus 800, 9700 AV, Groningen, Netherlands

    Aad Dijksma & Heinz Langer

  2. Technische Universität Sektion Mathematik, Mommsenstrasse 13, DDR-8027, Dresden, Germany

    Henk de Snoo

Authors
  1. Aad Dijksma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Heinz Langer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Henk de Snoo
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Davor Butković Svetozar Kurepa Hrvoje Kraljević

Rights and permissions

Reprints and permissions

Copyright information

© 1987 Springer-Verlag

About this chapter

Cite this chapter

Dijksma, A., Langer, H., de Snoo, H. (1987). Unitary colligations in Krein spaces and their role in the extension theory of isometries and symmetric linear relations in Hilbert spaces. In: Butković, D., Kurepa, S., Kraljević, H. (eds) Functional Analysis II. Lecture Notes in Mathematics, vol 1242. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072441

Download citation

  • DOI: https://doi.org/10.1007/BFb0072441

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-17833-0

  • Online ISBN: 978-3-540-47876-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation