ON THE CONTAINMENT OF C0 BY SPACES OF COMPACT-OPERATORS

We show that if F is a complemented subspace of a B-space having a suitable unconditional Schauder decomposition, then c(o) embeds into the space K(E, F) of compact operators from a B-space E into F, provided there is a noncompact operator from E into F. Then we consider some consequences of this result, showing that, sometimes, the coincidence of the space L(E, F) of all operators from E into F with K(E, F) is a necessary condition for the validity of certain well known results about the structure of K(E, F) and L(E, F).