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L- and M-weak compactness of positive semi-compact operators. (English) Zbl 1202.47041
Let \(E,F\) be Banach lattices. A continuous operator \(T :E\rightarrow F \) is called semi-compact if for each \(\varepsilon > 0\) there exists \( 0\leq u \in F\) such that \(T(U) \subseteq [-u,u] +\varepsilon V\), where \(U,V\) are the closed unit balls of \(E\) and \(F\), respectively. A continuous operator \(T :E\rightarrow F \) is called M-weakly compact if \(\lim_n ||Tx_n|| = 0\) for every norm bounded disjoint sequence \((x_n)\) in \(E\). A continuous operator \(T:E\rightarrow F \) is called L-weakly compact if \(\lim_n \|y_n\| = 0 \) for each sequence \((y_n)\) in the solid hull of \(T(U)\).
Each M-weakly (L-weakly) compact operator is semi-compact; however, a semicompact operator is not necessarily M-weakly or called L-weakly compact. The following theorem gives a characterization when each semi-compact operator is L-weakly compact.
Theorem. Each semi-compact \(T :E\rightarrow F \) is L-weakly compact if and only if \(F\) has order continuous norm.
Similarly, on M-weak compactness of semi-compact operators, the authors prove:
Theorem. Assume \(F\) is \(\sigma\)-Dedekind complete. Then the following are equivalent:
(1) Each positive semi-compact operator \(T :E\rightarrow F \) is M-weakly compact.
(2) One of the following holds:
(a) \(E'\) and \(F\) have order continuous norms.
(b) \(E\) is finite-dimensional.

MSC:
47B60 Linear operators on ordered spaces
46B42 Banach lattices
47B07 Linear operators defined by compactness properties
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[1] Aliprantis, C.D., Burkinshaw, O.: Positive Operators. New York: Academic Press (1985) · Zbl 0608.47039
[2] Aqzzouz, B., Elbour, A.: Characterization of the order weak compactness of semi-compact operators, J. Math. Anal. Appl., 355(2) (2009), 541–547 · Zbl 1190.47038 · doi:10.1016/j.jmaa.2009.01.072
[3] Aqzzouz, B., Elbour, A., Hmichane, J.: The duality problem for the class of b-weakly compact operators, Positivity, 13(4) (2009), 683–692 · Zbl 1191.47024 · doi:10.1007/s11117-008-2288-6
[4] Aqzzouz, B., Elbour, A., Hmichane, J.: Some properties of the class of positive Dunford-Pettis operators, J. Math. Anal. Appl., 354(1) (2009), 295–300 · Zbl 1167.47033 · doi:10.1016/j.jmaa.2008.12.063
[5] Chen, Z.L., Wickstead, A. W.: L-weakly and M-weakly compact operators, Indag. Math. (N.S), 10(3) (1999), 321–336 · Zbl 1028.47028 · doi:10.1016/S0019-3577(99)80025-1
[6] Groenewegen, G., Meyer-Nieberg, P.: An elementary and unified approach to disjoint sequence theorems,Nederl. Akad.Wetensch. Indag. Math., 48(3) (1986), 313–317 · Zbl 0637.46006
[7] Luxemburg, W.A.J., Zaanen, A.C.: Riesz spaces I. Amsterdam London: North Holland (1971) · Zbl 0231.46014
[8] Meyer-Nieberg, P.: Banach Lattices. Berlin, Heideberg,New York: SpringerVerlag (1991) · Zbl 0743.46015
[9] Wickstead, A.W.: Conversesfor theDodds-Freminand Kalton-Saab Theorems, Math. Proc. Cambridge Philos. Soc., 120 (1996), 175–179 · Zbl 0872.47018 · doi:10.1017/S0305004100074752
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