A Banach space X has the alternative Dunford–Pettis property if for every weakly convergent sequences (x n ) → x in X and (x n *) → 0 in X* with ||x n || = ||x||= 1 we have (x n *(x n )) → 0. We get a characterization of certain operator spaces having the alternative Dunford–Pettis property. As a consequence of this result, if H is a Hilbert space we show that a closed subspace M of the compact operators on H has the alternative Dunford–Pettis property if, and only if, for any h ∈ H, the evaluation operators from M to H given by S ↦ Sh, S ↦ S t h are DP1 operators, that is, they apply weakly convergent sequences in the unit sphere whose limits are also in the unit sphere into norm convergent sequences. We also prove a characterization of certain closed subalgebras of K(H) having the alternative Dunford-Pettis property by assuming that the multiplication operators are DP1.
Positivity – Springer Journals
Published: Jan 1, 2005
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