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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.

47B60 Linear operators on ordered spaces
46B42 Banach lattices
47B07 Linear operators defined by compactness properties
Full Text: DOI
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