, Volume 24, Issue 1, pp 141–149 | Cite as

Some properties of almost L-weakly and almost M-weakly compact operators

  • Aziz ElbourEmail author
  • Farid Afkir
  • Mohammed Sabiri


In this paper, we investigate necessary and sufficient conditions under which compact operators between Banach lattices must be almost L-weakly compact (resp. almost M-weakly compact). Mainly, it is proved that if X is a non zero Banach space then every compact operator \(T{:}X\rightarrow E\) (resp. \(T{:}E\rightarrow X\)) is almost L-weakly compact (resp. almost M-weakly compact) if and only if the norm on E (resp. \(E^{\prime }\)) is order continuous. Moreover, we present some interesting connections between almost L-weakly compact and L-weakly compact operators (resp. almost M-weakly compact and M-weakly compact operators).


L-weakly compact operator M-weakly compact operator Almost L-weakly compact operator Almost M-weakly compact Order continuous norm Banach lattice 

Mathematics Subject Classification

46B07 46B42 47B50 



The authors would like to thank the referee(s) for helpful comments and suggestions on the first version of the paper.


  1. 1.
    Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Berlin (2006)CrossRefGoogle Scholar
  2. 2.
    Aqzzouz, B., Elbour, A., Wickstead, A.W.: Compactness of L-weakly and M-weakly compact operators on Banach lattices. Rend. Circ. Mat. Palermo (2) 60(1), 43–50 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bouras, K., Lhaimer, D., Moussa, M.: On the class of almost L-weakly and almost M-weakly compact operators. Positivity 22, 1433–1443 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, Z.L., Wickstead, A.W.: L-weakly and M-weakly compact operators. Indag. Math. N.S. 10, 321–336 (1999)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dodds, P.G., Fremlin, D.H.: Compact operators on Banach lattices. Isr. J. Math. 34, 287–320 (1979)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Meyer-Nieberg, P.: Banach Lattices. Universitext. Springer, Berlin (1991)CrossRefGoogle Scholar
  7. 7.
    Zaanen, A.C.: Riesz Spaces II. North Holland Publishing Company, Amsterdam (1983)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Sciences and TechnologiesMoulay Ismaïl UniversityErachidiaMorocco

Personalised recommendations