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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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