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Semi-compactness of positive Dunford-Pettis operators on Banach lattices. (English) Zbl 1152.47012
An operator $$T$$ from a Banach space $$E$$ into an ordered Banach space $$F$$ is called semicompact if for each $$\varepsilon > 0$$, there exists $$0\leq u$$ in $$F^+$$ such that $$T(B_E)\subseteq [-u, u ]+ \varepsilon B_E$$, where $$B_E$$ and $$B_F$$ are the closed unit balls of $$E$$ and $$F$$. An operator $$T$$ between Banach spaces $$E$$ and $$F$$ is called a Dunford-Pettis operator if $$T$$ maps each weakly compact subset of $$E$$ into a compact subset of $$F$$. A Dunford-Pettis operator need not be semicompact.
The authors give necessary and sufficient conditions for a regular Dunford-Pettis operator to be semicompact. They also study the converse of this result and give sufficient conditions for a semicompact operator to be Dunford-Pettis. They also study the relation between semicompact operators and weakly compact operators. They give many useful consequences of their results among which they recapture a theorem of Aliprantis and Burkinshaw on compactness of Dunford-Pettis operators and a result of Wickstead on compactness of an operator $$S$$ satisfying $$0\leq S\leq T$$, where $$T$$ is compact.

##### MSC:
 47B07 Linear operators defined by compactness properties 47B60 Linear operators on ordered spaces 46A40 Ordered topological linear spaces, vector lattices 46B40 Ordered normed spaces 46B42 Banach lattices
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##### References:
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