Opuscula Math. 36, no. 6 (2016), 717-734

Opuscula Mathematica

Eigenvalue estimates for operators with finitely many negative squares

Jussi Behrndt
Roland Möws
Carsten Trunk

Abstract. Let \(A\) and \(B\) be selfadjoint operators in a Krein space. Assume that the resolvent difference of \(A\) and \(B\) is of rank one and that the spectrum of \(A\) consists in some interval \(I\subset\mathbb{R}\) of isolated eigenvalues only. In the case that \(A\) is an operator with finitely many negative squares we prove sharp estimates on the number of eigenvalues of \(B\) in the interval \(I\). The general results are applied to singular indefinite Sturm-Liouville problems.

Keywords: selfadjoint operator, Krein space, finitely many negative squares, eigenvalue estimate, indefinite Sturm-Liouville operator.

Mathematics Subject Classification: 47A55, 47B50.

Full text (pdf)

  1. T.Ya. Azizov, I.S. Iokhvidov, Linear Operators in Spaces with an Indefinite Metric, John Wiley and Sons, Chichester, New York, 1989.
  2. J. Behrndt, P. Jonas, Boundary value problems with local generalized Nevanlinna functions in the boundary condition, Integral Equations Operator Theory 55 (2006), 453-475.
  3. J. Behrndt, L. Leben, F. Martínez Pería, R. Möws, C. Trunk, Sharp eigenvalue estimates for rank one perturbations of nonnegative operators in Krein spaces, J. Math. Anal. Appl. 439 (2016), 864-895.
  4. J. Behrndt, M. Malamud, H. Neidhardt, Finite rank perturbations, scattering matrices and inverse problems, Oper. Theory Adv. Appl. 198 (2009), 61-85.
  5. J. Behrndt, R. Möws, C. Trunk, On finite rank perturbations of selfadjoint operators in Krein spaces and eigenvalues in spectral gaps, Complex Anal. Oper. Theory 8 (2014), 925-936.
  6. J. Behrndt, C. Trunk, On the negative squares of indefinite Sturm-Liouville operators, J. Differential Equations 238 (2007), 491-519.
  7. C. Bennewitz, B.M. Brown, R. Weikard, Scattering and inverse scattering for a left-definite Sturm Liouville problem, J. Differential Equations 253 (2012), 2380-2419.
  8. P. Binding, P. Browne, Left definite Sturm-Liouville problems with eigenparameter dependent boundary conditions, Differential Integral Equations 12 (1999), 167-182.
  9. P. Binding, P. Browne, B. Watson, Inverse spectral problems for left-definite Sturm-Liouville equations with indefinite weight, J. Math. Anal. Appl. 271 (2002), 383-408.
  10. M.Sh. Birman, M.Z. Solomjak, Spectral Theory of Selfadjoint Operators in Hilbert Space, Kluwer, Dordrecht, 1987.
  11. J. Bognar, Indefinite Inner Product Spaces, Springer, 1974.
  12. B. Ćurgus, H. Langer, A Krein space approach to symmetric ordinary differential operators with an indefinite weight function, J. Differential Equations 79 (1989), 31-61.
  13. V.A. Derkach, On Weyl function and generalized resolvents of a Hermitian operator in a Krein space, Integral Equations Operator Theory 23 (1995), 387-415.
  14. V.A. Derkach, M.M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991), 1-95.
  15. V.A. Derkach, M.M. Malamud, The extension theory of Hermitian operators and the moment problem, J. Math. Sci. (N.Y.) 73 (1995), 141-242.
  16. I. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, vol. 18, American Mathematical Society, Providence, R.I. 1969.
  17. S. Hassi, H.S.V. de Snoo, H. Woracek, Some interpolation problems of Nevanlinna-Pick type. The Krein-Langer method, Oper. Theory Adv. Appl. 106 (1998), 201-216.
  18. T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, 1976.
  19. Q. Kong, M. Möller, H. Wu, A. Zettl, Indefinite Sturm-Liouville problems, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), 639-652.
  20. Q. Kong, H. Wu, A. Zettl, Left-definite Sturm-Liouville problems, J. Differential Equations 177 (2001), 1-26.
  21. Q. Kong, H. Wu, A. Zettl, Singular left-definite Sturm-Liouville problems, J. Differential Equations 206 (2004), 1-29.
  22. M.G. Krein, H. Langer, Über einige Fortsetzungsprobleme, die eng mit der Theorie hermitescher Operatoren im Raum \(\Pi_{\kappa}\) zusammenhängen, I. Einige Funktionenklassen und ihre Darstellungen, Math. Nachr. 77 (1977), 187-236.
  23. H. Langer, Spektraltheorie linearer Operatoren in J-Räumen und einige Anwendungen auf die Schar \(L(\lambda) = \lambda^2 + \lambda B + C\), Habilitationsschrift, Technische Universität Dresden, 1965.
  24. H. Langer, Spectral functions of definitizable operators in Krein spaces, [in:] Functional Analysis Proceedings of a Conference held at Dubrovnik, Yugoslavia, November 2-14, 1981, Lecture Notes in Mathematics, vol. 948, Springer (1982), 1-46.
  25. H. Langer, A. Markus, V. Matsaev, Locally definite operators in indefinite inner product spaces, Math. Ann. 308 (1997), 405-424.
  26. H. Langer, B. Textorius, On generalized resolvents and \(Q\)-functions of symmetric linear relations (subspaces) in Hilbert space, Pacific J. Math. 72 (1977), 135-165.
  27. A. Zettl, Sturm-Liouville Theory, American Mathematical Society, Providence, RI, 2005.
  • Jussi Behrndt
  • Technische Universität Graz, Institut für Numerische Mathematik, Steyrergasse 30, 8010 Graz, Austria
  • Roland Möws
  • Krossener Str. 17, D-10245 Berlin, Germany
  • Carsten Trunk
  • Technische Universität Ilmenau, Institut für Mathematik, Postfach 100565, D-98684 Ilmenau, Germany
  • Communicated by A. Shkalikov.
  • Received: 2016-04-16.
  • Revised: 2016-05-24.
  • Accepted: 2016-05-24.
  • Published online: 2016-10-29.
Opuscula Mathematica - cover

Cite this article as:
Jussi Behrndt, Roland Möws, Carsten Trunk, Eigenvalue estimates for operators with finitely many negative squares, Opuscula Math. 36, no. 6 (2016), 717-734, http://dx.doi.org/10.7494/OpMath.2016.36.6.717

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

We advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.