Structure of subspaces of the compact operators having the Dunford-Pettis property

The structure of the subspaces having the Dunford-Pettis property (DPP) is studied, where is the space of all compact operators on and . The following conditions are shown to be equivalent: (i) M has the DPP, (ii) M is isomorphic to a subspace of (iii) the sets and are relatively compact for all and . The equivalence between (i) and (iii) was recently proven in the case of arbitrary Hilbert spaces by Brown and ülger. It is also shown that (i) and (ii) are equivalent for subspaces . This result is optimal in the sense that for there is a DPP-subspace that fails to be isomorphic to a subspace of . Received January 9, 1998; in final form October 1, 1998