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

The structure of the subspaces M subset of K (l(P)) having the Dunford-Pettis property (DPP) is studied, where K(l(P)) is the space of all compact operators on l(P) and 1 < p < infinity. The following conditions are shown to be equivalent: (i) M has the DPP, (ii) M is isomorphic to a subspace of co (iii) the sets {Sx : S is an element of B-M} subset of l(P) and {S*x* : S is an element of B-M} subset of l(P)' are relatively compact for all x is an element of l(P) and x* is an element of l(P)'. The equivalence between (i) and (iii) was recently proven in the case of arbitrary Hilbert spaces by Brown and Ulger. It is also shown that (i) and (ii) are equivalent for subspaces M subset of K (l(P)' + . . . + l(Pk)). This result is optimal in the sense that for 1 < p < q < infinity there is a DPP-subspace M subset of K (l(q)(l(P))) that fails to be isomorphic to a subspace of c(0). Mathematics Subject Classification (1991): 46B20, 46B28, 47D25.