Spectra of L1-convolution operators acting on Lp-spaces of commutative hypergroups

We show that, for commutative hypergroups, the spectrum of all L-1-convolution operators on L-p is independent of p is an element of [1, infinity] exactly when the Plancherel measure is supported on the whole character space chi(b)(K), i.e., exactly when L-1(K) is symmetric and for every alpha is an element of (K)over cap Reiter's condition P-2 holds true. Furthermore, we explicitly determine the spectra sigma(p)(T-epsilon 1) for the family of Karlin-McGregor polynomial hypergroups, which demonstrate that in general the spectra might even be different for each p.