Continuity of Operators Intertwining with Convolution Operators

Let G be a locally compact abelian group, let μ be a bounded complex-valued Borel measure on G, and let Tμ be the corresponding convolution operator on L1(G). Let X be a Banach space and let S be a continuous linear operator on X. Then we show that every linear operator Φ: X→L1(G) such that ΦS=TμΦ is continuous if and only if the pair (S,Tμ) has no critical eigenvalue.