L-Dunford-Pettis and Almost L-Dunford-Pettis Sets in Dual Banach Lattices

Main Article Content

Halimeh Ardakani
Manijeh Salimi


Following the concept of L-limited sets in dual Banach spaces introduced by Salimi and Moshtaghioun, we introduce the concepts of L-Dunford-Pettis and almost L-Dunford-Pettis sets in dual Banach lattices and then by a class of operators on Banach lattices, so called disjoint Dunford-Pettis completely continuous operators, we characterize Banach lattices in which almost L-Dunford-Pettis subsets of their dual, coincide with L-Dunford-Pettis sets.

Article Details


  1. C. D. Aliprantis and O. Burkishaw, Locally Solid Riesz Spaces, Academic Press, New York, London, 1978.
  2. C. D. Aliprantis and O. Burkishaw, Positive Operators, Academic Press, New York, London, 1978.
  3. B. Aqzzouz and K. Bouras, Dunford-Pettis sets in Banach lattices, Acta Math. Univ. Comenianae, 81 (2012), 185-196.
  4. B. Aqzzouz and K. Bouras, L-sets and almost L- sets in Banach lattices, Quaest. Math., 36 (2013), 107-118.
  5. B. Aqzzouz and A. Elbour, Some characterizations of almost Dunford-Pettis operators and applications, J. Positivity 15 (2011), 369-380.
  6. G. Emmanuele, Banach spaces in which Dunford-Pettis sets are relatively compact, Arch. Math., 58 (1992), 477-485.
  7. G. Emmanuele, The reciprocal Dunford-Pettis property and projective tensor products, Math. Proc. Cambridge Philos. Soc., 109 (1992), 161-166.
  8. K.E. Fahri, N. Machrafi and M. Moussa, Banach Lattices with the Positive Dunford-Pettis Relatively Compact Property, Extracta Math., 80 (2015), 161-179.
  9. G. Groenewegen and P. Meyer-Nieberg, An elementary and unified approach to disjoint sequence theorems, Indag. Math., 48 (1986), 313-317.
  10. P. Meyer- Nieberg, Banach Lattices, Universitext, Springer- Verlag, Berlin, 1991.
  11. K. Musial, The weak Radon-Nikodym property in Banach spaces, Studia Math., 64 (1979), 151-173.
  12. M. Salimi and S. M. Moshtaghioun, A new class of Banach spaces and its relation with some geometric properties of Bancah spaces, Abstr. Appl. Anal., ID 212957, 2012.
  13. J. A. Sanchez, Positive Schur property in Banach lattices, Extraccta Math., 7 (1992), 161-163.
  14. Y. Wen, Ji. Chen, Characterizations of Banach spaces with relatively compact Dunford-Pettis sets, Adv. Math., to appear.
  15. W. Wnuk, On the dual positive Schur property in Banach lattices, J. Positivity, 17 (2013), 759-773.