### On the weak compactness of Weak* Dunford-Pettis operators on Banach lattices

#### Abstract

We characterize Banach lattices on which each positive weak* Dunford-Pettis operator is weakly (resp., M-weakly, resp., order weakly) compact. More precisely, we prove that if $F$ is a Banach lattice with order continuous norm, then each positive weak* Dunford-Pettis operator $T : E \longrightarrow F$ is weakly compact if, and only if, the norm of $E^{\prime}$ is order continuous or $F$ is reflexive. On the other hand, when the Banach lattice $F$ is Dedekind $\sigma$-complete, we show that every positive weak* Dunford-Pettis operator $T: E \longrightarrow F$ is M-weakly compact if, and only if, the norms of $E^{\prime}$ and $F$ are order continuous or $E$ is finite-dimensional.

#### Article information

Source
Adv. Oper. Theory, Volume 2, Number 3 (2017), 192-200.

Dates
Accepted: 17 March 2017
First available in Project Euclid: 4 December 2017

https://projecteuclid.org/euclid.aot/1512431670

Digital Object Identifier
doi:10.22034/aot.1612-1078

Mathematical Reviews number (MathSciNet)
MR3730048

Zentralblatt MATH identifier
1380.46012

#### Citation

Kamal, El Fahri; Jawad, H'michane; Abdelmonim, El Kaddouri; Moulay Othmane, Aboutafail. On the weak compactness of Weak* Dunford-Pettis operators on Banach lattices. Adv. Oper. Theory 2 (2017), no. 3, 192--200. doi:10.22034/aot.1612-1078. https://projecteuclid.org/euclid.aot/1512431670

#### References

• C. D. Aliprantis and O. Burkinshaw, Positive operators, Reprint of the 1985 original, Springer, Dordrecht, 2006.
• B. Aqzzouz, R. Nouira, and L. Zraoula, Sur les opérateurs de Dunford-Pettis positifs qui sont faiblement compacts, (French) [On weakly compact positive Dunford-Pettis operators], Proc. Amer. Math. Soc. 134 (2006), 1161–1165.
• B. Aqzzouz, A. Elbour, and J. H'michane. On some properties of the class of semi-compact operators, Bull. Belg. Math. Soc. Simon Stevin 18 (2011), no. 4, 761–767.
• J. Borwein, M. Fabian, and J. Vanderwerff, Characterizations of Banach spaces via convex and other locally Lipschitz functions, Acta Math. Vietnam, 22 (1997), 53–69.
• N. Cheng, Z. L. Chen, and Y. Feng, L and M-weak compactness of positive semi-compact operators, Rend. Circ. Mat. Palermo 59 (2010), 101–105.
• J. X. Chen, Z. L. Chen, and G. X. Ji, Domination by positive weak* Dunford–Pettis operator on Banach lattices, Bull. Aust. Math. Soc. 90 (2014), 311–318.
• J. Diestel, Sequences and series in Banach spaces, Vol. 92 of Graduate Texts in Mathematics, Springer, Berlin, Germany, 1984.
• P. G. Dodds and D. H. Fremlin, Compact operators on Banach lattices, Israel J. Math. 34 (1979), 287–320.
• A. El Kaddouri, J. H'michane, K. Bouras, and M. Moussa, On the class of weak* Dunford–Pettis operators, Rend. Circ. Mat. Palermo (2) 62 (2013), 261–265.
• P. Meyer-Nieberg, Banach lattices, Universitext. Springer-Verlag, Berlin, 1991.
• H. H. Schaefer, Banach lattices and positive operators, Springer-Verlag, Berlin and New York, 1974.
• A. W. Wickstead, Converses for the Dodds-Fremlin and Kalton-Saab theorems, Math. Proc. Cambidge Philos. Soc. 120 (1996), 175–179.