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arXiv · 2411.17654

Testing compactness of linear operators

Timo S. Hänninen·Tuomas V. Oikari
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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.

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Timo S. Hänninen, Tuomas V. Oikari. 2024-11-26. Testing compactness of linear operators. https://arxiv.org/abs/2411.17654

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