A NOTE ON THE CONTINUITY OF OPERATORS INTERTWINING WITH CONVOLUTION OPERATORS

Abstract

Let $G$ be a MAP group, let $\mu$ be a bounded complex-valued Borel measure on $G$, and let $T_{\mu}$ be the corresponding convolution operator on $L^{1}(G)$. Let $X$ be a Banach space and let $S$ be a continuous linear operator on $X$. We show that every linear operator $\Phi:X \rightarrow L^{1}(G)$ such that $\Phi S = T_{\mu} \Phi$ is continuous if, and only if, the pair $(S,T_{\mu})$ has no critical eigenvalue.