Hypergroups of compact type
The aim of this paper is to study commutative hypergroups \(X\) for which the translation operator \(T_{x}-I\) is compact on the space \(L^{2}(X,m)\) (resp. \(L^{1}(X,m)\)) for each \(x\in X\), where \(m\) is the Haar measure on \(X\). These hypergroups are called hypergroups of compact type (resp. strong compact type). The main results are as follows: Hypergroups of compact type are necessarily discrete. Hypergroups of strong compact type have the property that the support of the Plancherel measure coincides with the set of all bounded characters; this kind of hypergroups are necessarily of compact type and symmetric, i.e. every bounded character is self-adjoint. It is also shown that for the class of hypergroups of strong compact type the ``modified '' Reiter condition \(\mathcal{P}_{1}\), introduced in [\textit{F. Filbir, R. Lasser, R. Szwarc}, Monatsh. Math. 143, 189-203 (2004; Zbl 1059.43005)] is satisfied for each nontrivial character. Finally, a full description of polynomial hypergroups of compact type (or equivalently in this particular case, of strong compact type) is presented and examples, such as the polynomial hypergroups generated by the little \(q\)-Legendre polynomials, are given.
scientific article; zbMATH DE number 2157669