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-Pett is property (DPP) is studied, where K(l(P)) is the space of all compact ope rators 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 i s 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 B rown and Ulger. It is also shown that (i) and (ii) are equivalent for subsp aces M subset of K (l(P)' + . . . + l(Pk)). This result is optimal in the s ense 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 S ubject Classification (1991): 46B20, 46B28, 47D25.