Nouira, Redouane; Lhaimer, Driss; Elbour, Aziz Some results on almost L-weakly and almost M-weakly compact operators. (English) Zbl 07511790 Positivity 26, No. 2, Paper No. 39, 11 p. (2022). Summary: The class of weakly compact operators does not contain that of almost L-weakly compact operators. In this paper, we provide a complete answer by giving necessary and sufficient conditions for which every positive almost L-weakly compact operator \(T:E\rightarrow F\) between two Banach lattices is weakly compact. On the other hand, we investigate conditions under which the adjoint operator of every positive almost L-weakly compact operator is almost M-weakly compact. Cited in 1 Document MSC: 47B60 Linear operators on ordered spaces 47B65 Positive linear operators and order-bounded operators 46B42 Banach lattices Keywords:almost L-weakly compact operator; almost M-weakly compact operator; weakly compact operator; Banach lattice; order continuous norm; KB-space PDFBibTeX XMLCite \textit{R. Nouira} et al., Positivity 26, No. 2, Paper No. 39, 11 p. (2022; Zbl 07511790) Full Text: DOI References: [1] Aliprantis, CD; Burkinshaw, O., Positive Operators (2006), Berlin: Springer, Berlin · Zbl 1098.47001 · doi:10.1007/978-1-4020-5008-4 [2] Afkir, F.; Bouras, K.; Elbour, A.; El Filali, S., Weak compactness of almost L-weakly and almost M-weakly compact operators, Quaest. Math., 44, 9, 1145-1154 (2021) · Zbl 1489.46025 · doi:10.2989/16073606.2020.1777482 [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) · Zbl 1496.46005 · doi:10.1007/s11117-018-0586-1 [4] Elbour, A.; Afkir, F.; Sabiri, M., Some properties of almost L-weakly and almost M-weakly compact operators, Positivity, 24, 141-149 (2020) · Zbl 1441.47021 · doi:10.1007/s11117-019-00671-7 [5] Aqzzouz, B.; Elbour, A.; Hmichane, J., The duality problem for the class of \(b\)-weakly compact operators, Positivity, 13, 683-692 (2009) · Zbl 1191.47024 · doi:10.1007/s11117-008-2288-6 [6] Meyer-Nieberg, P., Banach Lattices (1991), Berlin: Springer, Berlin · Zbl 0743.46015 · doi:10.1007/978-3-642-76724-1 [7] Meyer-Nieberg, P., Uber Klassen Schwach Kompakter Operatoren in Banachverbanden, Math. Z., 138, 145-159 (1974) · Zbl 0291.47020 · doi:10.1007/BF01214230 [8] Wnuk, W., Banach Lattices with Order Continuous Norms (1999), Warszawa: Polish Scientific Publishers PWN, Warszawa · Zbl 0948.46017 [9] Zaanen, AC, Riesz Spaces II (1983), Amsterdam: North Holland Publishing Company, Amsterdam · Zbl 0519.46001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.