Skip to main content
SpringerLink
Log in

Some characterizations of almost limited operators

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

In this paper we give several characterizations of almost limited operators. Mainly, it is proved that an operator \(T:X\rightarrow E\) from a Banach space X into a \(\sigma \)-Dedekind complete Banach lattice E is almost limited if and only if \(\left\| T^{*}\left( f_{n}\right) \right\| \rightarrow 0\) for every positive weak\(^{*}\) null sequence \(\left( f_{n}\right) \) of \(E^{*}\). Moreover, we present some interesting connections between almost limited, almost Dunford–Pettis and limited operators.

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. Aliprantis, C.D., Burkinshaw, O.: Positive operators. Springer, Berlin (2006)

    Book  MATH  Google Scholar 

  2. Aqzzouz, B., Elbour, A.: Some characterizations of almost Dunford–Pettis operators and applications. Positivity 15, 369–380 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aqzzouz, B., Elbour, A., Wickstead, A.W.: Positive almost Dunford–Pettis operators and their duality. Positivity 15, 185–197 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bourgain, J., Diestel, J.: Limited operators and strict cosingularity. Math. Nachr. 119, 55–58 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, J.X., Chen, Z.L., Ji, G.X.: Almost limited sets in Banach lattices. J. Math. Anal. Appl. 412, 547–553 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dodds, P.G.: o-Weakly compact mappings of Riesz spaces. Trans. Amer. Math. Soc. 214, 389–402 (1975)

    MathSciNet  MATH  Google Scholar 

  7. El Fahri, K., H’michane, J.: On the product of almost Dunford–Pettis and order weakly compact operators. Complex Anal. Oper. Theory 10(3), 605–615 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Meyer-Nieberg, P.: Banach lattices, Universitext. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  9. Wnuk, W.: A characterization of discrete Banach lattices with order continuous norms. Proc. Am. Math. Soc. 104, 197–200 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wnuk, W.: Banach lattices with order continuous norms. Polish Scientific Publishers PWN, Warsaw (1999)

    MATH  Google Scholar 

  11. Wnuk, W.: On the dual positive Schur property in Banach lattices. Positivity 17, 759–773 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zaanen, A.C.: Riesz Spaces II. North Holland Publishing Co., Amsterdam (1983)

    MATH  Google Scholar 

Download references

Acknowledgments

The author would like to thank the referee for helpful comments and suggestions on the first version of the paper.

Author information

Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences et Techniques, Université Moulay Ismaïl, B.P. 509, Errachidia, Morocco

    Aziz Elbour

Authors
  1. Aziz Elbour
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aziz Elbour.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Elbour, A. Some characterizations of almost limited operators. Positivity 21, 865–874 (2017). https://doi.org/10.1007/s11117-016-0437-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-016-0437-x

Keywords

Mathematics Subject Classification

Search

Navigation