Translation-finite sets, and weakly compact derivations from $\lp{1}(\Z_+)$ to its dual

TL;DR

This paper characterizes weakly compact derivations from the convolution algebra lat^1(a5_+) to its dual using combinatorial translation-finite sets, providing examples and exploring their relation to other smallness notions.

Contribution

It introduces a combinatorial characterization of weakly compact derivations via translation-finite sets and analyzes their properties and relationships to other smallness concepts.

Findings

Weakly compact derivations are characterized by translation-finite sets.

Examples of weakly compact but not compact derivations are provided.

Sets with positive Banach density are not translation-finite.

Abstract

We characterize those derivations from the convolution algebra $\ell^1({\mathbb Z}_+)$ to its dual which are weakly compact. In particular, we provide examples which are weakly compact but not compact. The characterization is combinatorial, in terms of "translation-finite" subsets of ${\mathbb Z}_+$, and we investigate how this notion relates to other notions of "smallness" for infinite subsets of ${\mathbb Z}_+$. In particular, we show that a set of strictly positive Banach density cannot be translation-finite; the proof has a Ramsey-theoretic flavour.