Skip to main content
SpringerLink
Log in

Partially Fundamentally Reducible Operators in Kreĭn Spaces

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

A self-adjoint operator A in a Kreĭn space (\({\mathcal{K}, [\cdot , \cdot]}\)) is called partially fundamentally reducible if there exist a fundamental decomposition \({\mathcal{K} = \mathcal{K}_{+}[\dot{+}]\mathcal{K}_{-}}\) (which does not reduce A) and densely defined symmetric operators S + and S − in the Hilbert spaces (\({\mathcal{K}_+, [\cdot , \cdot]}\)) and \({(\mathcal{K}_-, -[\cdot , \cdot])}\), respectively, such that each S + and S − has defect numbers (1, 1) and the operator A is a self-adjoint extension of \({S = S_{+} \oplus (-S_-)}\) in the Kreĭn space \({(\mathcal{K}, [\cdot , \cdot])}\). The operator A is interpreted as a coupling of operators S + and −S − relative to some boundary triples \({\big(\mathbb{C},\,\Gamma_0^+,\,\Gamma_1^+\big)}\) and \({\big(\mathbb{C},\,\Gamma_0^-,\,\Gamma_1^-\big)}\). Sufficient conditions for a nonnegative partially fundamentally reducible operator A to be similar to a self-adjoint operator in a Hilbert space are given in terms of the Weyl functions m + and m − of S + and S − relative to the boundary triples \({\big(\mathbb{C},\,\Gamma_0^+,\,\Gamma_1^+\big)}\) and \({\big(\mathbb{C},\,\Gamma_0^-,\Gamma_1^-\big)}\). Moreover, it is shown that under some asymptotic assumptions on m + and m − all positive self-adjoint extensions of the operator S are similar to self-adjoint operators in a Hilbert space.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akhiezer N.I., Glazman I.M.: Theory of Linear Operators in Hilbert Space, Two Volumes Bound as One. Dover Publications, New York (1993)

    Google Scholar 

  2. Akhiezer N.I., Glazman I.M.: Theory of Linear Operators in Hilbert Space. (Russian) vol. II, Third edition. Vishcha Shkola, Kharkov (1978)

    Google Scholar 

  3. Alpay D., Gohberg I.: Pairs of selfadjoint operators and their invariants. St. Petersburg Math. J. 16, 59–104 (2005)

    Article  MathSciNet  Google Scholar 

  4. Azizov T.Y., Behrndt J., Trunk C.: On finite rank perturbations of definitizable operators. J. Math. Anal. Appl. 339, 1161–1168 (2008)

    Article  MathSciNet  Google Scholar 

  5. Azizov T.Y., Iokhvidov I.S.: Linear Operators in Spaces with an Indefinite Metric. John Wiley & Sons, London (1990)

    Google Scholar 

  6. Bayasgalan, T.: On the fundamental reducibility of positive operators in spaces with indefinite metric. (Russian) Studia Sci. Math. Hungar. 13 (1978), 143–150 (1981)

  7. Behrndt J.: On the spectral theory of singular indefinite Sturm-Liouville operators. J. Math. Anal. Appl. 334, 1439–1449 (2007)

    Article  MathSciNet  Google Scholar 

  8. Behrndt J., Hassi S., de Snoo H., Wietsma R., Winkler H.: Linear fractional transformations of Nevanlinna functions associated with a nonnegative operator. Complex Anal. Oper. Theory 7, 331–362 (2013)

    Article  MathSciNet  Google Scholar 

  9. Behrndt J., Philipp F.: Spectral analysis of singular ordinary differential operators with indefinite weights. J. Differ. Equ. 248, 2015–2037 (2010)

    Article  MathSciNet  Google Scholar 

  10. Bennewitz C.: Spectral asymptotics for Sturm-Liouville equations. Proc. London Math. Soc. 59, 294–338 (1989)

    Article  MathSciNet  Google Scholar 

  11. Bognar J.: Indefinite Inner Product Spaces. Springer, New York (1974)

    Book  Google Scholar 

  12. Carmichael R.D.: Asymptotic analysis for complex-valued Stieltjes transforms. Integr. Transforms Spec. Funct. 22, 277–282 (2011)

    Article  Google Scholar 

  13. Carmichael R.D., Hayashi E.K.: Abelian theorems for the Stieltjes transform of functions. II. Internat. J. Math. Math. Sci. 4, 67–88 (1981)

    Article  MathSciNet  Google Scholar 

  14. Coddington E.A., Levinson N.: Theory of Ordinary Differential Equations. McGraw-Hill Book Company, New York (1955)

    Google Scholar 

  15. Ćurgus B.: On the regularity of the critical point infinity of definitizable operators. Integr. Equ Oper. Theory 8, 462–488 (1985)

    Article  Google Scholar 

  16. Ćurgus B., Dijksma A., Read T.: The linearization of boundary eigenvalue problems and reproducing kernel Hilbert spaces. Linear Algebra Appl. 329, 97–136 (2001)

    Article  MathSciNet  Google Scholar 

  17. Ćurgus B., Fleige A., Kostenko A.: The Riesz basis property of an indefinite Sturm-Liouville problem with non-separated boundary conditions. Integr. Equ. Oper. Theory 77, 533–557 (2013)

    Article  Google Scholar 

  18. Ćurgus B., Langer H.: A Kreĭn space approach to symmetric ordinary differential operators with an indefinite weight function. J. Differ. Equ. 79, 31–61 (1989)

    Article  Google Scholar 

  19. Ćurgus B., Najman B.: The operator \({({\rm sgn}\, x){d^{2}}/{dx^{2}}}\) is similar to a self-adjoint operator in \({L^{2}({\mathbb{R}})}\). Proc. Am. Math. Soc. 123, 1125–1128 (1995)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Derkach V.A., Hassi S., Malamud M.M., de Snoo H.S.V.: Generalized resolvents of symmetric operators and admissibility. Methods Funct. Anal. Topology 6, 24–55 (2000)

    MathSciNet  Google Scholar 

  24. Donoghue W.F.: Monotone Matrix Functions and Analytic Continuation Die Grundlehren der mathematischen Wissenschaften, Band 207. Springer-Verlag, Berlin (1974)

    Book  Google Scholar 

  25. Everitt W.N.: On a property of the m-coefficient of a second-order linear differential equation. J. London Math. Soc. 4(2), 443–457 (1971/1972)

  26. Everitt W.N., Zettl A.: On a class of integral inequalities. J. London Math. Soc. 17(2), 291–303 (1978/1979)

  27. Faddeev, M.M., Shterenberg, R.G.: On the similarity of some singular differential operators to selfadjoint operators. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 270 (2000, Russian), Issled. po Linein. Oper. i Teor. Funkts. 28, 336–349, 370–371; translation in J. Math. Sci. (N. Y.) 115 (2003) 2279–2286

  28. Fleige A.: A counterexample to completeness properties for indefinite Sturm-Liouville problems. Math. Nachr. 190, 123–128 (1998)

    Article  MathSciNet  Google Scholar 

  29. Fleige A., Najman B.: Nosingularity of Critical Points of Some Differential and Difference Operators, Oper. Theory: Adv. Appl., vol. 102. Birkhäuser, Basel (1998)

    Google Scholar 

  30. Garling D.J.H.: Inequalities: A Journey into Linear Analysis. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  31. Glazman, I.M.: Direct methods of qualitative spectral analysis of singular differential operators, Israel Program for Scientific Translations (1965)

  32. Gorbachuk V.I., Gorbachuk M.L.: Boundary Value Problems for Operator Differential Equations. Kluwer Academic Publishers Group, The Netherlands (1991)

    Book  Google Scholar 

  33. Horn R.A., Johnson C.R.: Matrix Analysis. 2nd edn. Cambridge University Press, Cambridge (2013)

    Google Scholar 

  34. Jonas, P.: Regularity criteria for critical points od definitizable operators. Oper. Theory Adv. Appl. 14, 179–195 (1984)

  35. Kac I.S., Kreĭn M.G.: R-functions–analytic functions mapping the upper halfplane into itself. Am. Math. Soc. Transl. Ser. 103(2), 1–18 (1974)

    Google Scholar 

  36. Karabash I.M.: J-selfadjoint ordinary differential operators similar to selfadjoint operators. Methods Funct. Anal. Topology 6, 22–49 (2000)

    MathSciNet  Google Scholar 

  37. Karabash, I.M.: A functional model, eigenvalues, and finite singular critical points for indefinite Sturm-Liouville operators. In: Topics in Operator Theory, vol. 2. Systems and Mathematical Physics, pp. 247–287, Oper. Theory Adv. Appl., 203. Birkhäuser Verlag, Basel, (2010)

  38. Karabash I.M., Kostenko A.: Indefinite Sturm-Liouville operators with the singular critical point zero. Proc. Roy. Soc. Edinburgh Sect. A 138, 801–820 (2008)

    Article  MathSciNet  Google Scholar 

  39. Karabash I.M., Kostenko A., Malamud M.M.: The similarity problem for J-nonnegative Sturm-Liouville operators. J. Differ. Equ. 246, 964–997 (2009)

    Article  MathSciNet  Google Scholar 

  40. Karabash I.M., Malamud M.M.: Indefinite Sturm-Liouville operators with finite zone potentials. Oper. Matrices 1, 301–368 (2007)

    Article  MathSciNet  Google Scholar 

  41. Karamata J.: Neuer Beweis und Verallgemeinerung der Tauberschen Sätze, welche die Laplacesche und Stieltjessche Transformation betreffen. J. Reine Angew. Math. 164, 27–39 (1931)

    MathSciNet  Google Scholar 

  42. Kochubei A.N.: Extensions of J-symmetric operators. (Russian) Teor. Funkciĭ Funkcional. Anal. Prilozhen. 31, 74–80 (1979)

    MathSciNet  Google Scholar 

  43. Kostenko A.S.: A spectral analysis of some indefinite differential operators. Methods Funct. Anal. Topology 12, 157–169 (2006)

    MathSciNet  Google Scholar 

  44. Kostenko A.: The similarity problem for indefinite Sturm-Liouville operators and the HELP inequality. Adv. Math. 246, 368–413 (2013)

    Article  MathSciNet  Google Scholar 

  45. Kostenko, A.: A note on J-positive block operator matrices. Integr. Equ. Oper. Theory 30 (2014)

  46. Kreĭn M.G.: The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications. I, (Russian) Mat. Sbornik N.S. 20(62), 431–495 (1947)

    Google Scholar 

  47. Kreĭn M.G., Langer H.: Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren in Raume \({\prod_{\kappa}}\) zusammenhängen. Teil I: Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77, 187– (1977)

    Google Scholar 

  48. Kuzhel S., Trunk C.: On a class of J-self-adjoint operators with empty resolvent set. J. Math. Anal. Appl. 379, 272–289 (2011)

    Article  MathSciNet  Google Scholar 

  49. Langer H.: Verallgemeinerte Resolventen eines J-nichtnegativen Operators mit endlichem Defekt. J. Funct. Anal. 8, 287–320 (1971)

    Article  Google Scholar 

  50. Langer, H.: Spectral functions of definitizable operators in Kreĭn spaces. In: Functional Analysis (Dubrovnik, 1981), Lecture Notes in Math. vol. 948, pp. 1–46. Springer, New York (1982)

  51. 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  Google Scholar 

  52. Levitan B.M.: On the asymptotic behavior of the spectral function of a self-adjoint differential equation of the second order. (Russian) Izvestiya Akad. Nauk SSSR. Ser. Mat. 16, 325–352 (1952)

    MathSciNet  Google Scholar 

  53. McEnnis B.W.: Fundamental reducibility of selfadjoint operators on Kreĭn space. J. Oper. Theory 8, 219–225 (1982)

    MathSciNet  Google Scholar 

  54. Read T.T.: A limit-point criterion for expressions with oscillatory coefficients. Pacific J. Math. 66, 243–255 (1976)

    Article  MathSciNet  Google Scholar 

  55. Veselić, K.: On spectral properties of a class of J-selfadjoint operators. I, II. Glasnik Mat. Ser. III 7(27) 229–248, (1972, ibid. 7(27):(1972) 249–254)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Western Washington University, 516 High Street, Bellingham, WA, 98226, USA

    Branko Ćurgus

  2. Department of Mathematics, Donetsk National University, 600-Richchya Str 21, Vinnytsya, 21021, Ukraine

    Vladimir Derkach

  3. Department of Mathematics, National Pedagogical University, Pirogova Str 9, Kiev, 01601, Ukraine

    Vladimir Derkach

Authors
  1. Branko Ćurgus
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vladimir Derkach
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Branko Ćurgus.

Additional information

The research of the second author was supported by the Fulbright Fund.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ćurgus, B., Derkach, V. Partially Fundamentally Reducible Operators in Kreĭn Spaces. Integr. Equ. Oper. Theory 82, 469–518 (2015). https://doi.org/10.1007/s00020-014-2204-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00020-014-2204-3

Mathematics Subject Classification

Keywords

Search

Navigation