On the class of \(U\)-Dunford-Pettis operators. (English) Zbl 1509.47055
An operator \(T: E \rightarrow Y\) between normed vector lattice \(E\) and Banach space \(Y\) is called \(U\)-Dunford-Pettis if \( T(x_{n}) \rightarrow 0\) in norm for every order bounded weak null sequence \((x_{n})\) in \(E\). In this paper, the authors show that the class of \(U\)-Dunford-Pettis operators coincides with that of order weakly compact operators when the lattice operations of Banach lattice \(E\) are weak sequentially continuous, and coincides with that of \(AM\)-compact operators when the Banach lattice \(E\) has order continuous norm. Also, they investigate the domination property of positive \(U\)-Dunford-Pettis operators and \(AM\)-compact operators.
Reviewer: Ömer Gök (İstanbul)
MSC:
| 47B60 | Linear operators on ordered spaces |
| 47B65 | Positive linear operators and order-bounded operators |
| 46B42 | Banach lattices |
Keywords:
Dunford-Pettis operator; \(AM\)-compact operator; \(U\)-Dunford-Pettis operator; order continuous norm; order unitReferences:
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