Skip to main content
SpringerLink
Log in

Locally Definitizable Operators: The Local Structure of the Spectrum

  • Reference work entry
  • First Online:
Operator Theory

Abstract

Locally definitizable operators have locally the same spectral properties as definitizable operators in Kreĭn spaces. It is shown in this note how to define spectral points of positive/negative type and spectral points of type \(\pi _{+}/\pi _{-}\) via approximative eigensequences. This approach has the advantage that it does not make use of a local spectral function. Moreover, perturbation results for locally definitizable operators are discussed. Spectral points of type π + and π − are stable under compact perturbations. For real spectral points of type π + and type π − which are not in the interior of the spectrum the growth of the resolvent in an open neighborhood of these spectral points is of finite order. This can be utilized to show the existence of a local spectral function with singularities. With the help of this local spectral function one can also characterize spectral points of positive/negative type and spectral points of type π + and type π −: It turns out that all spectral subspaces corresponding to sufficiently small neighborhoods of spectral points of positive/negative type are Hilbert or anti-Hilbert spaces and spectral subspaces corresponding to spectral points of type π + or type π − are Pontryagin spaces.

Locally definitizable operators are used in the study of indefinite Sturm–Liouville problems, λ-dependent boundary value problems, \(\mathcal{P}\mathcal{T}\)-symmetric operators, and partial differential equations and in the study of problems of Klein–Gordon type.

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. Adamjan, V., Langer, H., Möller, M.: Compact perturbation of definite type spectra of self-adjoint quadratic operator pencils. Integr. Equ. Oper. Theory 39, 127–152 (2001)

    Article  MATH  Google Scholar 

  2. Arens, R.: Operational calculus of linear relations. Pac. J. Math. 11, 9–23 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  3. Azizov, T.Ya., Iokhvidov, I.S.: Linear Operators in Spaces with an Indefinite Metric. Wiley, Chichester (1989)

    Google Scholar 

  4. Azizov, T.Ya., Jonas, P.: On compact perturbations of normal operators in a Krein space. Ukrainskii Matem. Zurnal 42, 1299–1306 (1990)

    MathSciNet  Google Scholar 

  5. Azizov, T.Ya., Strauss, V.A.: Spectral decompositions for special classes of self-adjoint and normal operators on Krein spaces. In: Spectral Theory and Applications. Theta Ser. Adv. Math., vol. 2, pp. 45–67. Theta, Bucharest (2003)

    Google Scholar 

  6. Azizov, T.Ya., Trunk, C.: On domains of \(\mathcal{P}\mathcal{T}\) symmetric operators related to \(-y''(x) + (-1)^{n}x^{2n}y(x)\). J. Phys. A Math. Theory 43, 175303 (2010)

    Google Scholar 

  7. Azizov, T.Ya., Trunk, C.: \(\mathcal{P}\mathcal{T}\) symmetric, Hermitian and \(\mathcal{P}\)-self-adjoint operators related to potentials in \(\mathcal{P}\mathcal{T}\) Quantum Mechanics. J. Math. Phys. 53, 012109 (2012)

    Google Scholar 

  8. Azizov, T.Ya., Jonas, P., Trunk, C.: Spectral points of type π + and π − of selfadjoint operators in Krein spaces. J. Funct. Anal. 226, 114–137 (2005)

    Google Scholar 

  9. Azizov, T.Ya., Behrndt, J., Jonas, P., Trunk, C.: Compact and finite rank perturbations of linear relations in Hilbert spaces. Integr. Equ. Oper. Theory 63, 151–163 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Azizov, T.Ya., Jonas, P., Trunk, C.: Small perturbation of selfadjoint and unitary operators in Krein spaces. J. Oper. Theory 64, 401–416 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Azizov, T.Ya., Behrndt, J., Jonas, P., Trunk, C.: Spectral points of type π + and type π − for closed linear relations in Krein spaces. J. Lond. Math. Soc. 83, 768–788 (2011)

    Google Scholar 

  12. Bayasgalan, Ts.: Fundamental reducibility of normal operators on Krein space. Stud. Sci. Math. Hung. 35, 147–150 (1999)

    MathSciNet  MATH  Google Scholar 

  13. Behrndt, J.: A class of abstract boundary value problems with locally definitizable functions in the boundary condition. Integr. Equ. Oper. Theory 55, 453–475 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Behrndt, J.: Finite rank perturbations of locally definitizable self-adjoint operators in Krein spaces. J. Oper. Theory 58, 415–440 (2007)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Behrndt, J.: Boundary value problems with eigenvalue depending boundary conditions. Math. Nachr. 282, 659–689 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Behrndt, J.: Spectral theory of elliptic differential operators with indefinite weights. Proc. R. Soc. Edinb. Sect. A 143, 21–38 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Behrndt, J., Jonas, P.: On compact perturbations of locally definitizable selfadjoint relations in Krein spaces. Integr. Equ. Oper. Theory 52, 17–44 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Behrndt, J., Jonas, P.: Boundary value problems with local generalized Nevanlinna functions in the boundary condition. Integr. Equ. Oper. Theory 55, 453–475 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Behrndt, J., Trunk, C.: Sturm-Liouville operators with indefinite weight functions and eigenvalue depending boundary conditions. J. Differ. Equ. 222, 297–324 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Behrndt, J., Trunk, C.: On the negative squares of indefinite Sturm-Liouville operators. J. Differ. Equ. 238, 491–519 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Behrndt, J., Philipp, F., Trunk, C.: Properties of the spectrum of type π + and type π − of self-adjoint operators in Krein spaces. Methods Funct. Anal. Topol. 12, 326–340 (2006)

    MathSciNet  MATH  Google Scholar 

  24. 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 

  25. Behrndt, J., Katatbeh, Q., Trunk, C.: Accumulation of complex eigenvalues of indefinite Sturm-Liouville operators, J. Phys. A Math. Theory 41, 244003 (2008)

    Article  MathSciNet  Google Scholar 

  26. Behrndt, J., Katatbeh, Q., Trunk, C.: Non-real eigenvalues of singular indefinite Sturm-Liouville operators. Proc. Am. Math. Soc. 137, 3797–3806 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Behrndt, J., Möws, R., Trunk, C.: Eigenvalue estimates for singular left-definite Sturm-Liouville operators. J. Spectr. Theory 1, 327–347 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Behrndt, J., Luger, A., Trunk, C.: On the negative squares of a class of self-adjoint extensions in Krein spaces. Math. Nachr. 286, 118–148 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Behrndt, J., Möws, R., Trunk, C.: On finite rank perturbations of selfadjoint operators in Krein spaces and eigenvalues in spectral gaps. Complex Anal. Oper. Theory 8, 925–936 (2013)

    Article  Google Scholar 

  30. Behrndt, J., Philipp, F., Trunk, C.: Bounds on the non-real spectrum of differential operators with indefinite weights. Math. Ann. 357, 185–213 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Behrndt, J., Leben, L., Martínez Pería, F., Möws, R., Trunk, C.: Sharp eigenvalue estimates for rank one perturbations of nonnegative operators in Krein spaces, Preprint/Technische Universität Ilmenau, Institut für Mathematik; 2013-13

    Google Scholar 

  32. Bender, C.M., Brody, D.C., Jones, H.F.: Complex extension of quantum mechanics. Phys. Rev. Lett. 89, 270401 (2002)

    Article  MathSciNet  Google Scholar 

  33. Bender, C.M., Brody, D.C., Jones, H.F.: Must a Hamiltonian be Hermitian? Am. J. Phys. 71, 1095 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Binding, P., Möller, M.: Negativity indices for definite and indefinite Sturm-Liouville problems. Math. Nachr. 283, 180–192 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Bognar, J.: Indefinite Inner Product Spaces. Springer, Berlin (1974)

    Book  MATH  Google Scholar 

  36. Caliceti, E., Graffi, S., Sjöstrand, J.: Spectra of \(\mathcal{P}\mathcal{T}\)-symmetric operators and perturbation theory. J. Phys. A Math. Gen. 38, 185–193 (2005)

    Article  MATH  Google Scholar 

  37. Cross, R.: Multivalued Linear Operators. Monographs and Textbooks in Pure and Applied Mathematics, vol. 213. Marcel Dekker, New York (1998)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  39. Ćurgus, B., Najman, B.: Positive differential operators in Krein space \(L^{2}(\mathbb{R})\). Oper. Theory Adv. Appl. 87, 95–104 (1996)

    Google Scholar 

  40. Ćurgus, B., Najman, B.: Preservation of the range under perturbation of an operator. Proc. Am. Math. Soc. 125, 2627–2631 (1997)

    Article  MATH  Google Scholar 

  41. Daho, K., Langer, H.: Sturm-Liouville operators with an indefinite weight function. Proc. R. Soc. Edinb. Sect. A 78, 161–191 (1977/78)

    Google Scholar 

  42. Dijksma, A., de Snoo, H.S.V.: Symmetric and selfadjoint relations in Krein Spaces I. Oper. Theory Adv. Appl. 24, 145–166 (1987)

    Google Scholar 

  43. Dijksma, A., de Snoo, H.S.V.: Symmetric and selfadjoint relations in Krein Spaces II. Ann. Acad. Sci. Fenn. Math. 12, 199–216 (1987)

    Article  MATH  Google Scholar 

  44. Dijksma, A., Langer, H.: Operator theory and ordinary differential operators. In: Lectures on Operator Theory and Its Applications. Fields Inst. Monogr. vol. 3. American Mathematical Society, Providence (1996)

    Google Scholar 

  45. Dritschel, M.: Compact perturbations of operators on Krein spaces. In: Harmonic Analysis and Operator Theory. Contemporary Mathematics, vol. 189, pp. 201–211. American Mathematical Society, Providence (1995)

    Google Scholar 

  46. Fleige, A.: Spectral Theory of Indefinite Krein-Feller Differential Operators. Mathematical Research, vol. 98. Akademie, Berlin (1996)

    Google Scholar 

  47. Günther, U., Stefani, F., Znojil, M.: MHD α 2-dynamo, squire equation and \(\mathcal{P}\mathcal{T}\)-symmetric interpolation between square well and harmonic oscillator. J. Math. Phys. 46, 063504 (2005)

    Article  MathSciNet  Google Scholar 

  48. Haase, M.: The Functional Calculus for Sectorial Operators. Operator Theory: Advances and Applications, vol. 169. Birkhäuser, Basel (2006)

    Google Scholar 

  49. Iohvidov, I.S., Krein, M.G., Langer, H.: Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric. Akademie, Berlin (1982)

    MATH  Google Scholar 

  50. Jacob, B., Trunk, C., Winklmeier, M.: Analyticity and Riesz basis property of semigroups associated to damped vibrations. J. Evol. Equ. 8, 263–281 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  51. Jonas, P.: Compact perturbations of definitizable operators. II. J. Oper. Theory 8, 3–18 (1982)

    MathSciNet  MATH  Google Scholar 

  52. Jonas, P.: On a class of unitary operators in Krein space. Oper. Theory Adv. Appl. 17, 151–172 (1986)

    MathSciNet  Google Scholar 

  53. Jonas, P.: On a class of selfadjoint operators in Krein space and their compact perturbations. Integr. Equ. Oper. Theory 11, 351–384 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  54. Jonas, P.: A note on perturbations of selfadjoint operators in Krein spaces. Oper. Theory Adv. Appl. 43, 229–235 (1990)

    MathSciNet  Google Scholar 

  55. Jonas, P.: On a problem of the perturbation theory of selfadjoint operators in Krein spaces. J. Oper. Theory 25, 183–211 (1991)

    MathSciNet  MATH  Google Scholar 

  56. Jonas, P.: On the spectral theory of operators associated with perturbed Klein-Gordon and wave type equation. J. Oper. Theory 29, 207–224 (1993)

    MathSciNet  MATH  Google Scholar 

  57. Jonas, P.: Riggings and relatively form bounded perturbations of nonnegative operators in Krein spaces. Oper. Theory Adv. Appl. 106, 259–273 (1998)

    MathSciNet  Google Scholar 

  58. Jonas, P.: On bounded perturbations of operators of Klein-Gordon type. Glas. Mat. 35, 59–74 (2000)

    MathSciNet  MATH  Google Scholar 

  59. Jonas, P.: On locally definite operators in Krein spaces. In: Spectral Theory and Applications. Theta Ser. Adv. Math. vol. 2, pp. 95–127. Theta, Bucharest (2003)

    Google Scholar 

  60. Jonas, P.: On operator representations of locally definitizable functions. Oper. Theory Adv. Appl. 162, 165–190 (2005)

    Article  MathSciNet  Google Scholar 

  61. Jonas, P., Langer, H.: Compact perturbations of definitizable operators. J. Oper. Theory 2, 63–77 (1979)

    MathSciNet  MATH  Google Scholar 

  62. Jonas, P., Trunk, C.: On a class of analytic operator functions and their linearizations. Math. Nachr. 243, 92–133 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  63. Jonas, P., Trunk, C.: A Sturm-Liouville problem depending rationally on the eigenvalue parameter. Math. Nachr. 280, 1709–1726 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  64. Karabash, I., Trunk, C.: Spectral properties of singular Sturm-Liouville operators. Proc. R. Soc. Edinb. Sect. A 139, 483–503 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  65. Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  66. Kong, Q., Wu, H., Zettl, A., Möller, M.: Indefinite Sturm-Liouville problems. Proc. R. Soc. Edinb. Sect. A 133, 639–652 (2003)

    Article  MATH  Google Scholar 

  67. Krein, M.G.: Introduction to the theory of indefinite J-spaces and to the theory of operators in those spaces. Am. Math. Soc. Transl. 93, 103–176 (1970)

    MathSciNet  Google Scholar 

  68. Krein, M.G., Langer, H.: On some mathematical principles in the linear theory of damped oscillations of continua I. Integr. Equ. Oper. Theory 1, 364–399 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  69. Krein, M.G., Langer, H.: On some mathematical principles in the linear theory of damped oscillations of continua II. Integr. Equ. Oper. Theory 1, 539–566 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  70. Lancaster, P., Markus, A.S., Matsaev, V.I.: Definitizable operators and quasihyperbolic operator polynomials. J. Funct. Anal. 131, 1–28 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  71. Lancaster, P., Shkalikov, A.A., Ye, Q.: Strongly definitizable linear pencils in Hilbert space. Integr. Equ. Oper. Theory 17, 338–360 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  72. Langer, H.: Spektraltheorie linearer Operatoren in J-Räumen and einige Anwendungen auf die Schar \(L(\lambda ) =\lambda ^{2}I +\lambda B + C\). Habilitationsschrift, Technische Universität Dresden (1965)

    Google Scholar 

  73. Langer, H.: Spektralfunktionen einer Klasse J-selbstadjungierter Operatoren. Math. Nachr. 33, 107–120 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  74. Langer, H.: Über stark gedämpfte Scharen im Hilbertraum (German). J. Math. Mech. 17, 685–705 (1968)

    MathSciNet  MATH  Google Scholar 

  75. Langer, H.: Über eine Klasse polynomialer Scharen selbstadjungierter Operatoren im Hilbertraum (German). J. Funct. Anal. 12, 13–29 (1973)

    Article  MATH  Google Scholar 

  76. Langer, H.: Über eine Klasse polynomialer Scharen selbstadjungierter Operatoren im Hilbertraum. II (German). J. Funct. Anal. 16, 221–234 (1974)

    Google Scholar 

  77. Langer, H.: Spectral functions of definitizable operators in Krein spaces. In: Functional Analysis (Dubrovnik, 1981). Lecture Notes in Mathematics, vol. 948, pp. 1–46 (1982)

    Article  Google Scholar 

  78. Langer, H., Szafraniec, F.H.: Bounded normal operators in Pontryagin spaces. Oper. Theory Adv. Appl. 162, 231–251 (2006)

    Article  MathSciNet  Google Scholar 

  79. Langer, H., Tretter, C.: A Krein space approach to \(\mathcal{P}\mathcal{T}\)-symmetry. Czech. J. Phys. 54, 1113–1120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  80. Langer, H., Mennicken, R., Möller, M.: A second order differential operator depending nonlinearly on the eigenvalue parameter. Oper. Theory Adv. Appl. 48, 319–332 (1990)

    Google Scholar 

  81. Langer, H., Markus, A.S., Matsaev, V.I.: Locally definite operators in indefinite inner product spaces. Math. Ann. 308, 405–424 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  82. Langer, H., Markus, A.S., Matsaev, V.I.: Linearization and compact perturbation of self-adjoint analytic operator functions. Oper. Theory Adv. Appl. 118, 255–285 (2000)

    MathSciNet  Google Scholar 

  83. Langer, H., Markus, A.S., Matsaev, V.I.: Self-adjoint analytic operator functions and their local spectral function. J. Funct. Anal. 235, 193–225 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  84. Langer, H., Langer, M., Markus, A.S., Tretter, C.: The Virozub-Matsaev Condition and spectrum of definite type for self-adjoint operator functions. Complex Anal. Oper. Theory 2, 99–134 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  85. Langer, H., Najman, B., Tretter, C.: Spectral theory of the Klein-Gordon equation in Krein spaces. Proc. Edinb. Math. Soc., II. 51, 711–750 (2008)

    Google Scholar 

  86. Lyubich, Yu.I., Matsaev, V.I.: On operators with decomposable spectrum. Am. Math. Soc. Transl. II. Ser. 47, 89–129 (1965); translation from Mat. Sb., N. Ser. 56(98), 433–468 (1962)

    Google Scholar 

  87. Markus, A.S.: Introduction to the Spectral Theory of Operator Polynomials. Translation of Mathematical Monographs, vol. 71. American Mathematical Society, Providence (1988)

    Google Scholar 

  88. Shkalikov, A.A.: Dissipative operators in the Krein space. Invariant subspaces and properties of restrictions. Funct. Anal. Appl. 41, 154–167 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  89. Trunk, C.: Spectral theory for operator matrices related to models in mechanics. Math. Notes 83, 843–850 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  90. Trunk, C.: Analyticity of semigroups related to a class of block operator matrices. Oper. Theory Adv. Appl. 195, 257–271 (2009)

    MathSciNet  Google Scholar 

  91. Veselić, K.: On spectral properties of a class of J-selfadjoint operators, I. Glasnik Matematički 7, 229–247 (1972)

    Google Scholar 

  92. Veselić, K.: On spectral properties of a class of J-selfadjoint operators, II. Glasnik Matematički 7, 249–254 (1972)

    Google Scholar 

  93. Zettl, A.: Sturm-Liouville Theory. Mathematical Surveys and Monographs, vol. 121. American Mathematical Society, Providence (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Technische Universität Ilmenau, Ilmenau, Germany

    Carsten Trunk

Authors
  1. Carsten Trunk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carsten Trunk .

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

Trunk, C. (2015). Locally Definitizable Operators: The Local Structure of the Spectrum. In: Alpay, D. (eds) Operator Theory. Springer, Basel. https://doi.org/10.1007/978-3-0348-0667-1_38

Download citation

Publish with us

Policies and ethics

Search

Navigation