Remarks on compactness of operators defined on

Abstract

This note presents several observations on Banach spaces X such that, for fixed , every operator from an -space into X which is weakly compact is already compact.The interest in such objects is due to the fact that a Banach space X has the above property for if and only if, for some and then all , every strictly q-integral operator with values in X is already q-integral. Recall that a Banach space X has the Radon-Nikodym property iff every strictly 1 -integral X-valued operator is nuclear. We shall, however, not discuss any Radon-Nikodym aspects here;these can be found in C. Cardassi's theory [3].