Remarks on compactness of operators defined on $Lp$

This note presents several observations on Banach spaces X such that, for fixed $1 ≤ p ≤ ∈fty$, every operator from an $Lp$-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 $2≤ p &lt;∈fty$ if and only if, for some and then all $2 ≤ q < ∈fty$, 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]