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Translation-finite sets and weakly compact derivations from \(\ell^{1}(\mathbb Z_{+})\) to its dual. (English) Zbl 1204.43002
The weak compactness of derivations from the convolution algebra \(l^1(\mathbb{Z}_+)\) to its dual is studied. Every bounded derivation from \(l^1(\mathbb{Z}_+)\) to its dual is of the form
\[ D_\psi(\delta_0)=0\quad\text{and}\quad D_\psi(\delta_j)(\delta_k) =\frac{j}{j+k}\,\psi_{j+k}\quad (j\in\mathbb{N},\,k\in\mathbb{Z}_+) \]
for some \(\psi\in l^\infty(\mathbb{N})\). A set \(S\subset\mathbb{Z}_+\) is called translation-finite (TF) if for every sequence \(n_1<n_2<\dots\) in \(\mathbb{Z}_+\) there exists \(k\in\mathbb{N}\) such that \(\bigcap_{i=1}^k(S-n_i)\) is finite or empty.
Main result: Given \(\psi\in l^\infty(\mathbb{N})\), the derivation \(D_\psi\) is weakly compact if and only if for all \(\varepsilon>0\) the set \(S_\varepsilon=\{n\in\mathbb{N}:|\psi_n|>\varepsilon\}\) is translation-finite.
Explicit examples of non-compact derivations \(D_\psi\) that are weakly compact are presented. The TF property is then compared with other notions of size. Combinatorics of TF subsets of \(\mathbb{Z}_+\) is considered as well.

MSC:
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
47B07 Linear operators defined by compactness properties
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