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 $$\underset{i\to \infty }{\lim }\underset{f_i\in F_i}{\sup }\Vert T{f}_i{\Vert }_Y=0$$ 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 the literature and by characterizing the $L^p\to L^p$ compactness of dyadic paraproducts on general measure spaces through introducing a space of vanishing mean oscillations with respect to mass (as opposed to length) of cubes.

arXiv comment29 pages. Added a characterization of when the masswise and lengthwise VMO spaces coincide, expanded the introduction, and made minor edits