Skip to main content
SpringerLink
Log in

Quantitative Characterizations of Bounded Compact Approximation Property by Semi-Fredholm Operators

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

Based on a characterization of upper semi-Fredholm operators due to A. Lebow and M. Schechter, we introduce and investigate a new quantity characterizing upper semi-Fredholm operators. This quantity and several well-known quantities are used to characterize bounded compact approximation property. Similarly, a new quantity characterizing lower semi-Fredholm operators is introduced, investigated and used to characterize the bounded compact approximation property for dual spaces.

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. Albiac, F., Kalton, N. J.: Topics in Banach Space Theory, Springer, New York, 2006

    MATH  Google Scholar 

  2. Astala, K., Tylli, H.-O.: On the bounded compact approximation property and measures of noncompactness. J. Funct. Anal., 70, 388–401 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Axler, S., Jewell, N., Shields, A.: The essential norm of an operator and its adjoint. Trans. Amer. Math. Soc., 261, 159–167 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  4. Casazza, P. G.: Approximation properties, In: Handbook of the Geometry of Banach Spaces, Vol.1, W.B. Johnson and J. Lindenstrauss, eds., Elsevier, Amsterdam, 2001, 271–316

    Chapter  Google Scholar 

  5. Chen, D., Ruan, Y.: Quantitative Bessaga—Pełczyński property and quantitative Rosenthal property. Math. Nachr., 292, 1685–1700 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  6. Faĭnshteĭn, A. S.: Measures of noncompactness of linear operators and analogoues of the minimum modulus for semi-Fredholm operators, In: Spectral Theory of Operators and its Applications, No. 6, “Èlm”, Baku, 1985, 182–195

    Google Scholar 

  7. Galaz-Fontes, F.: Measures of noncompactness and upper semi-Fredholm perturbation theorems. Proc. Amer. Math. Soc., 118, 891–897 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gol’denšteĭn, L. S., Gokhberg, I. Ts., Markus, A. S.: Investigation of some properties of bounded linear operators in connection with their q-norms. Uch. Zap. Kishinev. Gos. Univ., 29, 29–36 (1957)

    Google Scholar 

  9. Gol’denšteĭn, L. S., Markus, A. S.: On the measure of non-compactness of bounded sets and of linear operators (in Russian), In: Studies in Algebra and Math. Anal., Izdat. Karta Moldovenjaske, Kishinev, 1965, 45–54

    Google Scholar 

  10. González, M., Martinón, A.: Operational quantities characterizing semi-Fredholm operators. Studia Math., 114, 13–27 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. González, M., Martinón, A.: Quantities characterizing semi-Fredholm operators and perturbation radii. J. Math. Anal. Appl., 390, 362–367 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kato, T.: Perturbation theory for nullity deficiency and other quantities of linear operators. J. Anal. Math., 6, 261–322 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lebow, A., Schechter, M.: Semigroups of operators and measures of noncompactness. J. Funct. Anal., 7, 1–26 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lima, Å., Nygaard, O., Oja, E.: Isometric factorization of weakly compact operators and the approximation property. Israel J. Math., 119, 325–348 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I, Sequence Spaces, Springer, Berlin, 1977

    MATH  Google Scholar 

  16. Martinón, A.: Distortion of Banach spaces and supermultiplicative operational quantities. J. Math. Anal. Appl., 363, 655–662 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pietsch, A.: Operator ideals, Deutsch. Verlag Wiss., Berlin, 1978, North-Holland Publishing Company, Amsterdam/New York/Oxford, 1980

    MATH  Google Scholar 

  18. Schechter, M.: Quantities related to strictly singular operators. Indiana Univ. Math. J., 21, 1061–1071 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  19. Schechter, M., Whitley, R.: Best Fredholm perturbation theorems. Studia Math., 90, 175–190 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tylli, H.-O.: On the asymptotic behavior of some quantities related to semi-Fredholm operators. J. London Math. Soc., 31, 340–348 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  21. Yood, B.: Properties of linear transformations preserved under addition of a completely continuous transformation. Duke Math. J., 18, 599–612 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zemánek, J.: Geometric characteristics of semi-Fredholm operators and their asymptotic behaviour. Studia Math., 80, 219–234 (1984)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the anonymous referees for useful suggestions.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, P. R. China

    Dong Yang Chen

  2. School of Mathematics and Statistics, Fujian Normal University, Fuzhou, 350007, P. R. China

    Ying Bin Ruan

Authors
  1. Dong Yang Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ying Bin Ruan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dong Yang Chen.

Additional information

The first author was supported by the National Natural Science Foundation of China (Grant No. 11971403) and the Natural Science Foundation of Fujian Province of China (Grant No. 2019J01024)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, D.Y., Ruan, Y.B. Quantitative Characterizations of Bounded Compact Approximation Property by Semi-Fredholm Operators. Acta. Math. Sin.-English Ser. 39, 1115–1128 (2023). https://doi.org/10.1007/s10114-023-1007-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-023-1007-3

Keywords

MR(2010) Subject Classification

Search

Navigation