Testing compactness of linear operators

Abstract: Let $(F_i)$ be a sequence of sets in a Banach space $X$. For what sequences does the condition $$ \limsup_{i\to \infty} \sup_{f_i\in F_i} |Tf_i|_Y=0 $$ hold for every Banach space $Y$ and every compact operator $T:X\to Y$? We answer this question by giving sufficient (and necessary) criteria for such sequences. We illustrate the applicability of the criteria by examples from literature and by characterizing the $L^p\to L^p$ compactness of dyadic paraproducts on general measure spaces.