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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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##### References:
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