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Banach spaces in which Dunford-Pettis sets are relatively compact. (English) Zbl 0761.46010

This paper is concerned with the study of the class of Banach spaces in which Dunford-Pettis sets are relatively compact (in symbols (DPrcP)). The interest in this class of spaces is due to the following result: \(E^*\) has the (DPrcP) iff \(E\) does not contain \(\ell^ 1\). This fact is then used to show that if \(E\), \(F\) do not contain \(\ell^ 1\) and \(L(E,F^*)=K(E,F^*)\), then \(E\otimes F\) does not contain \(\ell^ 1\), so answering a question by Ruess. The above family of Banach spaces is also utilized to prove that if \(E\) has the Dunford-Pettis property and \(F\) the (DPrcP) then every dominated operator from \(C(K,E)\) into \(F\) is a Dunford-Pettis operator (we note that if \(E\) is finite dimensional this is true for all \(F\), but if \(E\) is infinite dimensional this can be false). At the end, several results showing that special Banach lattices as well as spaces of compact operators have the (DPrcP) are presented.

MSC:

46B28 Spaces of operators; tensor products; approximation properties
46B22 Radon-Nikodým, Kreĭn-Milman and related properties
47L05 Linear spaces of operators
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