Skip to main content
SpringerLink
Log in

Generalized Orlicz-Lorentz sequence spaces and corresponding operator ideals

  • Published:
Mathematica Slovaca

Abstract

In this paper we introduce generalized or vector-valued Orlicz-Lorentz sequence spaces l p,q,M (X) on Banach space X with the help of an Orlicz function M and for different positive indices p and q. We study their structural properties and investigate cross and topological duals of these spaces. Moreover these spaces are generalizations of vector-valued Orlicz sequence spaces l M (X) for p = q and also Lorentz sequence spaces for M(x) = x q for q ≥ 1. Lastly we prove that the operator ideals defined with the help of scalar valued sequence spaces l p,q,M and additive s-numbers are quasi-Banach operator ideals for p < q and Banach operator ideals for p ≥ q. The results of this paper are more general than the work of earlier mathematicians, say A. Pietsch, M. Kato, L. R. Acharya, etc.

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. ACHARYA, L. R.: Linear Operators and Approximation Quantites. Dissertation, Indian Institute of Technology, Kanpur, 2008.

    Google Scholar 

  2. CARL, B.— STEPHANI, I.: Entropy, Compactness and the Approximation of Operators, Cambridge Univ. Press, Cambridge, 1990.

    Book  MATH  Google Scholar 

  3. FORALWESKI, P.— HUDZIK, H.— SZYMASZKIEWICZ, L.: On some geometric and topological prpperties of generalized Orlicz-Lorentz sequence spaces, Math. Nachr. 281 (2008), 181–198..

    Article  MathSciNet  Google Scholar 

  4. GARLING, D. J. H.: On symmetric sequence spaces I, Proc. Lond. Math. Soc. (3) 16 (1966), 85–106.

    Article  MATH  MathSciNet  Google Scholar 

  5. GREGORY, D. A.: Vector-valued Sequence Spaces. Dissertation, Univ. of Michigan, Ann. Arbor, 1967.

    Google Scholar 

  6. GROTHENDIECK, A.: Sur une notion de produit tensoriel topologique d’espaces vectoriels topologiques, et une, classes remarquable d’espaces liees a cette notion, C. R. Math. Acad. Sci. Paris 233 (1951), 1556–1558.

    MATH  MathSciNet  Google Scholar 

  7. GUPTA MANJUL— PRADHAN SHESADEV: On certain type of modular sequence spaces, Turkish J. Math. 32 (2008), 293–303.

    MATH  MathSciNet  Google Scholar 

  8. LINDENSTRAUSS, J.— TZAFRIRI, L.: Classical Banach Spaces I, Sequence Spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1977.

    Book  MATH  Google Scholar 

  9. KAMTHAN, P. K.— GUPTA, M.: Sequence Spaces and Series. Lect. Notes Pure Appl. Math. 65, Marcel Dekker, Inc., New York-Basel, 1981.

    MATH  Google Scholar 

  10. KATO, M.: On Lorentz Spaces l p,q{E}, HiroshimaMath. J. 6 (1976), 73–93.

    MATH  Google Scholar 

  11. DE GRANDE-KIMPE, M.: Generalized sequence spaces, Bull. Soc. Math. Belgique XXIII (1971).

  12. KRASNOSELSKII, M. A.— RUTISKY, Y. B.: Convex Functions and Orlicz Spaces, Groningen, 1961.

    Google Scholar 

  13. MUSIELAK, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Math. 1034, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1983.

    MATH  Google Scholar 

  14. PATTERSON, J.: Generalized Sequence Spaces and Matrix Transformations. Dissertation, I.I.T. Kanpur, India, 1980.

    Google Scholar 

  15. PIETSCH, A.: Zur Fredholmchen Theorie in lokalkonvexen Räumen, Studia Math. 22 (1963), 161–179.

    MATH  MathSciNet  Google Scholar 

  16. PIETSCH, A.: Einigeneue Klassen von kompakten linearen Abbildungen, Rev. Roumaine Math. Pures Appl. 8 (1963), 427–447.

    MATH  MathSciNet  Google Scholar 

  17. PIETSCH, A.: s-numbers of operators in Banach spaces, Studia Math. 51 (1974), 201–223.

    MATH  MathSciNet  Google Scholar 

  18. PIETSCH, A.: Operator Ideals, Math. Monograph. 16, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.

    Google Scholar 

  19. PIETSCH, A.: Eigenvalues and s-numbers, Cambridge Stud. Adv. Math. 13, Cambridge Univ. Press, Cambridge, 1987.

    MATH  Google Scholar 

  20. RUCKLE, W.: The construction of sequence spaces that have schauder bases, Canad. J. Math. 19 (1967), 828–838.

    Article  MATH  MathSciNet  Google Scholar 

  21. RUDIN, W.: Functional Analysis, McGraw-Hill, Inc., New York, 1976.

    Google Scholar 

  22. SARGENT, W. L. C: Some sequence spaces related to the lp spaces, J. Lond. Math. Soc. (2) 35 (1960), 161–171.

    Article  MATH  MathSciNet  Google Scholar 

  23. SCHMIDT, E.: Zur Theorie der linearen nichtlinearnen Integralgleichungen, Math. Ann. 63; 64 (1907), 433–476; 161–174.

    Article  Google Scholar 

  24. YILMAZ, Y.— KERMAL ÖZDEMIR, M.— SOLAK, I.— CANDAN, M.: Operator on some vector-valued Orlicz sequence spaces, Fen Derg. 17 (2005), 59–71.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India

    Manjul Gupta & Antara Bhar

Authors
  1. Manjul Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antara Bhar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manjul Gupta.

Additional information

Communicated by Lajos Molnar

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gupta, M., Bhar, A. Generalized Orlicz-Lorentz sequence spaces and corresponding operator ideals. Math. Slovaca 64, 1475–1496 (2014). https://doi.org/10.2478/s12175-014-0287-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s12175-014-0287-6

2010 Mathematics Subject Classification

Keywords

Search

Navigation