Testing compactness of linear operators

TL;DR

This paper characterizes sequences in Banach spaces that ensure the compactness condition for all compact operators, providing criteria and applications to dyadic paraproducts in measure spaces.

Contribution

It offers necessary and sufficient criteria for sequences in Banach spaces that guarantee the compactness condition for all compact operators, with applications to dyadic paraproducts.

Findings

Established criteria for sequences ensuring compactness conditions.

Applied criteria to characterize $L^p\to L^p$ compactness of dyadic paraproducts.

Provided examples illustrating the criteria's applicability.

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.