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About positive Dunford–Pettis operators on Banach lattices. (English) Zbl 1112.47028
The main purpose of the paper is to characterize Banach lattices on which each positive Dunford–Pettis operator is compact. The following theorem gives new sufficient conditions for this.
Theorem. Let \(E, F\) be Banach lattices. Each positive Dunford–Pettis operator from \(E\) into \(F\) is compact if one of the following holds: (i) \(E\) and its dual \(E'\) have order continuous norms. (ii) \(E'\) is discrete and has order continuous norm. (iii) \(E'\) has order continuous norm, \(F\) is discrete and has order continuous norm. (iv) \(F\) is finite-dimensional.
Examples are provided to show that the sufficient conditions of the above theorem are not necessary. In Theorem 2.12, a converse to the above result is given. Whenever the lattice operations in \(E\) are weakly sequentially continuous or the norm of \(F\) is order continuous, the sufficient conditions of the preceding theorem are shown to be necessary.

MSC:
47B65 Positive linear operators and order-bounded operators
47B07 Linear operators defined by compactness properties
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