Almost L-Dunford-Pettis sets in Banach lattices and its applications

Abstract

We introduce and study the notion of almost L-Dunford-Pettis sets in Banach lattices and we give some characterizations of it in terms of sequences. As an application, we establish new properties of almost Dunford-Pettis completely continuous operators. Finally, by introducing the concept of aL-Dunford-Pettis property in Banach lattices, we investigate the weak compactness of almost Dunford-Pettis completely continuous operator.

Keywords

• Banach lattice,Dunford-Pettis set,relatively compact Dunford-Pettis property,Dunford-Pettis completely continuous operator

References

1. [1] C.D. Aliprantis and O. Burkinshaw, Positive operators, Reprint of the 1985 original. Springer, Dordrecht, 2006.
2. [2] K.T. Andrews, Dunford-Pettis sets in the space of Bochner integrable functions, Math. Ann. 241, 35-41, 1979.
3. [3] K. Bouras, Almost Dunford-Pettis sets in Banach lattices, Rend. Circ. Mat. Palermo. 62, 227-236, 2013.
4. [4] K. El Fahri, N. Machrafi and M. Moussa, Banach Lattices with the Positive Dunford- Pettis Relatively Compact Property, Extracta Math. 30 (2), 161-179, 2015.
5. [5] P. Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991.
6. [6] A. Retbi and B. El Wahbi, L-Dunford-Pettis property in Banach spaces, Methods Funct. Anal. Topology, accepted.
7. [7] Y. Wen and J. Chen, Characterizations of Banach Spaces With Relatively Compact Dunford-Pettis Sets, Advances in Mathematics (China), 45 (1), 122-132, 2016.