Multipliers on \(L^p\)-spaces for hypergroups (Q370846)

The authors characterize multipliers on \(L^{p}(K)\) where \(K\) is a locally compact commutative hypergroup as the space of convolutors of \(L^{p}(K)\). A multiplier on \(L^{p}(K)\) is defined as a bounded linear operator that commutes with translation. When \(p=2\) (the case of \(L^{2}(K)\)), the space of multipliers is proved to be isometric to \(\mathcal{P}(K)\), the space of pseudomeasures on \(K\). Most of the results are extensions to the hypergroup case of \textit{R. Larsen}'s result [An introduction to the theory of multipliers. Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0213.13301)] for locally compact abelian groups. Unlike the group situation the dual space \(\hat{K}\) of a hypergroup \(K\) does not have a hypergroup structure, and even when it does the Plancherel measure \(\pi\) is not supported by the entire dual space. As a consequence a hypergroup translation operator is not always defined on \(S=\mathrm{supp}(\pi)\). In order to resolve this issue, a translation operator for \(L^{1}(S,\pi)\) is defined via the Plancherel transform, which is used to characterize multipliers on \(L^{1}(S,\pi)\) and \(L^{2}(S,\pi)\). As a special case the authors establish several results on multipliers on polynomial hypergroups, where \(K\) is the set of nonnegative integers, with the dual space being compact.