Positivity 10 (2006), 51–63 c  2006 Birkh¨auser Verlag Basel/Switzerland 1385-1292/010051-13 DOI 10.1007/s11117-005-0007-0 Positivity The Alternative Dunford–Pettis Property for Subspaces of the Compact Operators Mar´ ıa D. Acosta and Antonio M. Peralta Abstract. A Banach space X has the alternative Dunford–Pettis property if for every weakly convergent sequences (xn) → x in X and (x * n ) → 0 in X * with ‖xn‖ = ‖x‖ = 1 we have (x * n (xn)) → 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 sub- space 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. Keywords. The Dunford-Pettis property, the alternative Dunford-Pettis Prop- erty, compact operators. 1. Introduction A Banach space X has the Dunford-Pettis property (DP in the sequel) if for any Banach space Y , every weakly compact operator from X to Y is completely con- tinuous, that is, it maps weakly compact subsets of X onto norm compact subsets of Y . The DP was introduced by Grothendieck who also showed that a Banach space X has the DP if, and only if, for every weakly null sequences (x n ) in X and (x ∗ n ) in X ∗ we have x ∗ n (x n ) → 0. Since its introduction by Grothendieck, the DP has had an important development. We refer to Diestel [10] as an excellent survey on the DP and to Bourgain [2], Bunce [4], Chu-Iochum [7] and Talagrand [17] and Chum-Mellon [8] for more recent results. Recently, S. Brown and A. ¨ Ulger (see Brown [3] and ¨ Ulger [18]) have stud- ied the DP for subspaces of the compact operators on an arbitrary Hilbert space. Indeed, if M is a closed subspace of the compact operators on a Hilbert space H,