Skip to main content
Springer Nature Link
Log in

On the measure of noncompactness of linear operators in spaces of strongly α-summable and bounded sequences

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

In this paper, we characterize classes of matrix transformations from BK spaces into spaces of bounded sequences and their subclasses of infinite matrices that define compact operators. Furthermore, using these results and the solvability of certain infinite linear systems we give necessary and sufficient conditions for A to be a compact operator on spaces that are strongly α-bounded or summable.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. R. Labbas and B. de Malafosse, On some Banach algebra of infinite matrices and applications, Demonstratio Matematica, 31 (1998), 153–168.

    MATH  Google Scholar 

  2. I. J. Maddox, Infinite matrices of operators, Springer-Verlag, Berlin — Heidelberg — New York, 1980.

    MATH  Google Scholar 

  3. B. de Malafosse, Bases in sequence spaces and expansion of a function in a series of power series, Matematicki Vesnik, 52/3–4 (2000), 99–112.

    Google Scholar 

  4. B. de Malafosse, Application of the sum of operators in the commutative case to the infinite matrix theory, Soochow J. Math., 27 (2001), 405–421.

    MATH  MathSciNet  Google Scholar 

  5. B. de Malafosse, Properties of some sets of sequences and application to the spaces of bounded difference sequences of order µ, Hokkaido Math. J., 31 (2002), 283–299.

    MATH  MathSciNet  Google Scholar 

  6. B. de Malafosse, Recent results in the infinite matrix theory and application to Hill equation, Demonstratio Matematica, 35 (2002), 11–26.

    MATH  Google Scholar 

  7. B. de Malafosse, Variation of an element in the matrix of the first difference operator and matrix transformations, Novi Sad J. Math., 32 (2002), 141–158.

    MATH  Google Scholar 

  8. B. de Malafosse, Sets of sequences that are strongly τ-bounded and matrix transformations between these sets, Demonstratio Matematica, 36 (2003), 155–171.

    MATH  Google Scholar 

  9. B. de Malafosse, On some BK space, Int. J. Math. Math. Sci., 28 (2003), 1783–1801.

    Article  Google Scholar 

  10. B. de Malafosse and E. Malkowsky, Sequence spaces and inverse of an infinite matrix, Rend. del Circ. Mat. di Palermo Serie II, 51 (2002), 277–294.

    MATH  Google Scholar 

  11. E. Malkowsky, Linear operators in certain BK spaces, Approximation Theory and Function Series Budapest (Hungary), Bolyai Mathematical Studies 5, Budapest, 1996, 259–273.

  12. E. Malkowsky, Matrixabbildungen in paranormierten FK-Räumen, Analysis, 7 (1987), 275–292.

    MATH  MathSciNet  Google Scholar 

  13. E. Malkowsky, BK spaces, bases and linear operators, Rend. del Circ. Mat. di Palermo. Serie II. Suppl., 52 (1998), 177–191.

    MathSciNet  Google Scholar 

  14. E. Malkowsky and V. Rakočević, The measure of noncompactness of linear operators between certain sequence spaces, Acta Sci. Math. (Szeged), 64 (1998), 151–171.

    MATH  MathSciNet  Google Scholar 

  15. E. Malkowsky, V. Rakočević, An introduction into the theory of sequence spaces and measure of noncompactness, Zbornik radova, Matematički institut SANU, 9/17 (2000), 143–234.

    Google Scholar 

  16. F. Móricz, On Λ-strong convergence of numerical sequences and Fourier series, Acta Math. Hungar., 54 (1989), 319–327.

    Article  MathSciNet  Google Scholar 

  17. A. Peyerimhoff, Über ein Lemma von Herrn Chow, J. London Math. Soc., 32 (1957), 33–36.

    Article  MATH  MathSciNet  Google Scholar 

  18. A. Wilansky, Functional Analysis, Blaisdell Publishing Company, New York, 1964.

    MATH  Google Scholar 

  19. A. Wilansky, Summability through Functional Analysis, North-Holland Mathematics Studies 85, 1984.

Download references

Author information

Authors and Affiliations

  1. I.U.T Université du Havre, 76600, Le Havre, France

    Bruno de Malafosse

  2. Mathematisches Institut, Universität Giessen, Arndtstrasse 2, D-35392, Giessen, Germany

    Eberhard Malkowsky

  3. Department of Mathematics, Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, 18000, Niš, Yugoslavia

    Eberhard Malkowsky

Authors
  1. Bruno de Malafosse
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Eberhard Malkowsky
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Bruno de Malafosse.

Additional information

Communicated by Dénes Petz

Research is supported by the German DAAD foundation (German Academic Exchange Service), the University of Le Havre and the research project #1232 of the Serbian Ministry of Science, Technology and Development.

Rights and permissions

Reprints and permissions

About this article

Cite this article

de Malafosse, B., Malkowsky, E. On the measure of noncompactness of linear operators in spaces of strongly α-summable and bounded sequences. Period Math Hung 55, 129–148 (2007). https://doi.org/10.1007/s10998-007-4129-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-007-4129-4

Mathematics subject classification number

Key words and phrases