Spectra ofL1-convolution operators acting onLp-spaces of commutative hypergroups

We show that, for commutative hypergroups, the spectrum of allL1-convolution operators onLpis independent ofp∈ [1, ∞] exactly when the Plancherel measure is supported on the whole character space χb(K), i.e., exactly whenL1(K) is symmetric and for every α ∈Reiter's conditionP2holds true. Furthermore, we explicitly determine the spectra σp(Tϵ1) for the family of Karlin–McGregor polynomial hypergroups, which demonstrate that in general the spectra might even be different for eachp.