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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.

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
Full Text: DOI
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