OPERATORS INTERTWINING WITH CONVOLUTION OPERATORS ON HYPERGROUPS

Summary: Let G be a locally compact hypergroup and let K be a compact subhypergroup such that (G,K) is a Gelfand pair. Let μ be a bounded complex-valued Borel measure on G , and let T♮μ be the corresponding convolution operator of L1♮(G), the subset of L1(G) consisting of K-bi-invariant functions. Suppose that S is a bounded linear operator on a Banach space X. We prove that every linear operator Ψ : X −→ L1♮(G) such that ΨS = T♮ μΨ is continuous if and only if (S, T ♮ μ) has no critical eigenvalues.