Skip to main content

On regularizers of unbounded linear operators in Banach spaces

Regularizers of densely defined unbounded linear operators in Banach spaces and their applications to spectral theory are considered. Necessary and sufficient conditions in terms of regularizer properties for an unbounded operator T to have discrete spectrum are obtained. In the case where T has a self-adjoint regularizer in some Schatten--von Neumann ideals, asymptotic properties of the eigenvalues are investigated; in particular, it is shown that the eigenvalues of T asymptotically belong to some angle in the complex plane. Bibliography: 16 titles.


  1. S. G. Mikhlin, Multi-dimensional Singular Integrals and Integral Equations [in Russian], Moscow (1962).

  2. S. S. Kutateladze, Foundations of Functional Analysis [in Russian], Novosibirsk, Inst. Math. Publ. (200).

  3. S. Prössdorf, Some Classes of Singular Integral Equations [Russian translation], Moscow (1973).

  4. V. A. Solonnikov, ``On operators that admit regularization,'' Zap. Nauchn. Semin. LOMI, 21, 159--163 (1971).

    MathSciNet  MATH  Google Scholar 

  5. O. I. Panich and V. A. Solonnikov, ``On the question of an equivalent right regularizer,'' Zap. Nauchn. Semin. LOMI, 22, 192--195 (1971).

    Google Scholar 

  6. I. Ts. Gokhberg and M. G. Krein, Introduction to the Theory of Linear Non-self-adjoint Operators in a Hilbert Space [in Russian], Moscow (1965).

  7. A. S. Markus and V. I. Matsaev, ``Theorem on comparison of spectra and the spectral asymptotic for the M. V. Keldysh pencil,'' Mat. Sb., 123, 391--406 (1984).

    MathSciNet  Google Scholar 

  8. A. S. Markus and V. I. Matsaev, ``On the spectral asymptotic for operators that are close to normal ones,'' Funkts. Anal. Prilozh., 13, 93--94 (1979).

    MathSciNet  MATH  Google Scholar 

  9. A. S. Markus and V. I. Matsaev, ``Theorems on comparison of spectra of linear operators and spectral asymptotics,'' Trudy Mosk. Mat. Ob., 45, 133--181 (1982).

    MathSciNet  MATH  Google Scholar 

  10. G. V. Radzievskii, ``Asymptotic of distribution of characteristic numbers of operator-functions that are analytic in an angle,'' Mat. Sb., 112, 396--420 (1980).

    MathSciNet  Google Scholar 

  11. L. Hörmander, The Analysis of Linear Partial Differential Operators. III. Pseudo-Differential Operators} [Russian translation], Moscow (1987).

  12. V. M. Kaplitskii, ``On asymptotic distribution of eigenvalues for a self-adjoint hyperbolic differential operator of the second order on the two-dimensional torus,'' Sib. Mat. Zh., 51, 1041--1060 (2010).

    MathSciNet  Google Scholar 

  13. G. V. Rozenblyum, M. Z. Solomyak, and M. A. Shubin, Spectral Theory of Differential Operators, Itogi Nauki Tekhn., 64, VINITI, Moscow (1989).

  14. V. V. Kozlov and I. V. Volovich, ``Finite action Klein-Gordon solutions on Loretzian manifolds,'' Int. J. Geom. Meth. Mod. Phys., 3, 1349--1357 (2006).

    MathSciNet  Article  MATH  Google Scholar 

  15. A. S. Andreev, ``Asymptotics of spectra of compact pseudodifferential operators on a Euclidean set,'' Mat. Sb., 179, 202--223 (1988).

    Google Scholar 

  16. A. S. Andreev, ``Estimation of spectra of compact pseudodifferential operators in unbounded domains,'' Mat. Sb., 181, 995--1006 (1990).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to V. M. Kaplitskii.

Additional information

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 401, 2012, pp. 93–102.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kaplitskii, V.M. On regularizers of unbounded linear operators in Banach spaces. J Math Sci 194, 651–655 (2013).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI:


  • Banach Space
  • Linear Operator
  • Complex Plane
  • Spectral Theory
  • Asymptotic Property