## Annals of Functional Analysis

### Strong relatively compact Dunford–Pettis property in Banach lattices

#### Abstract

In this paper, the concept of the strong relatively compact Dunford–Pettis property (strong DP$_{\mathit{rc}}$P) in Banach lattices is introduced and Banach lattices with the strong DP$_{\mathit{rc}}$P are characterized. Next, by introducing the class of almost Dunford–Pettis completely continuous operators from an arbitrary Banach lattice $E$ to another $F$, we give some properties of them related to some well-known classes of operators and, especially, some related to the strong DP$_{\mathit{rc}}$P of the Banach lattice $E$.

#### Article information

Source
Ann. Funct. Anal., Volume 7, Number 2 (2016), 270-280.

Dates
Accepted: 2 August 2015
First available in Project Euclid: 29 February 2016

https://projecteuclid.org/euclid.afa/1456754404

Digital Object Identifier
doi:10.1215/20088752-3475634

Mathematical Reviews number (MathSciNet)
MR3652768

Zentralblatt MATH identifier
1348.46022

#### Citation

Ardakani, Halimeh; Modarres Mosadegh, S. M. Sadegh. Strong relatively compact Dunford–Pettis property in Banach lattices. Ann. Funct. Anal. 7 (2016), no. 2, 270--280. doi:10.1215/20088752-3475634. https://projecteuclid.org/euclid.afa/1456754404

#### References

• [1] C. D. Aliprantis and O. Burkishaw, Positive Operators, Academic Press, New York, 1978.
• [2] C. D. Aliprantis and O. Burkishaw, Locally Solid Riesz Spaces, Academic Press, New York, 1978.
• [3] B. Aqzzouz and K. Bouras, Dunford-Pettis sets in Banach lattices, Acta Math. Univ. Comenianae 81 (2012), no. 2, 185–196.
• [4] K. Bouras, Almost Dunford-Pettis sets in Banach lattices, Rend. Circ. Mat. Palermo 62 (2013), 227–236.
• [5] J. X. Chen, Z. L. Chen, and G. X. Ji, Almost limited sets in Banach lattices, J. Math. Anal. Appl. 412 (2014), no. 1, 547–563.
• [6] L. Drewnowski, On Banach spaces with the Gelfand-Phillips property, Math. Z. 193 (1986), 405–411.
• [7] G. Emmanuele, Banach spaces in which Dunford-Pettis sets are relatively compact, Arch. Math. 58 (1992), 477–485.
• [8] H. Z. Lin, The weakness of limited set and limited operator in Banach spaces, Pure Appl. Math 27 (2011), no. 5, 650–655.
• [9] P. Meyer-Nieberg, Banach Lattices, Universitext, Springer, Berlin, 1991.
• [10] N. Machrafi, A. Elbour, and M. Moussa, Some characterizations of almost limited sets and applications, preprint, arXiv:1312.2770v1 [math.FA].
• [11] K. Musial, The weak Radon-Nikodym property in Banach spaces, Studia Math. 64 (1979), 151–173.
• [12] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, I: Sequence Spaces, Ergeb. Math. Grenzgeb. (3) 92, Springer, Berlin, 1977.
• [13] Y. Wen and J. Chen, Characterizations of Banach spaces with relatively compact Dunford-Pettis sets, preprint, to appear in Adv. Math.
• [14] W. Wnuk, Some characterizations of the Banach lattices with the Schur property, Rev. Mat. Univ. Complutense Madr. 2 (1989), 217–224.
• [15] W. Wnuk, Banach lattices with the weak Dunford–Pettis property, Atti Semin. Mat. Fis. Univ. Modena 42 (1994), no. 1, 227–236.