The alternative Dunford–Pettis property for subspaces of the compact operators. (English) Zbl 1107.46014
A Banach space \(X\) is said to have the Dunford–Pettis property (DP) if for all weakly null sequences \(\{x_n\}_{n \geq 1} \subset X\) and \(\{x^\ast_n\}_{n \geq 1} \subset X^\ast\), \(x^\ast_n(x_n) \rightarrow 0\). In the present paper, the authors study a weaker version of this property called Alternative Dunford–Pettis property (DP1) introduced by W. Freedman [Stud.Math.125, 143–159 (1997; Zbl 0897.46009)]. A Banach space has the DP1 if one has the same conclusion as in DP for weakly convergent sequences in the unit sphere of \(X\). The authors characterize closed subspaces of compact operators on a Hilbert space in terms of evaluation operators being DP1. They also have similar results for closed subspaces of compact operators between reflexive Banach spaces with Schauder basis that satisfy certain additional conditions. They also give an example to show that results of this nature cannot be expected for general subspaces of operators.
Reviewer: T.S.S.R.K. Rao (Bangalore)
MSC:
| 46B28 | Spaces of operators; tensor products; approximation properties |
| 46B03 | Isomorphic theory (including renorming) of Banach spaces |
Citations:
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