Continuity of operators intertwining with convolution operators (Q1865325)

The authors give an affirmative solution to the following problem of \textit{Laursen} and \textit{Neumann} [An Introduction to Local Spectral Theory, London Mathematical Society Monographs, New Series, Vol. 20, Oxford University Press (2000; Zbl 0957.47004)]: Suppose that \(\mu\) and \(\nu\) are bounded complex valued measures on a locally compact abelian group \(G\) such that the corresponding pair \((T_\nu,T_\mu)\) of convolution operators on \(L^1(G)\) has no critical eigenvalue. Is every linear operator \(\Phi:L^1(G) \to L^1(G)\) such that \(T_\nu\circ \Phi=\Phi \circ T_\mu\) automatically continuous? This is a generalization of a question of Johnson from 1967. The main theorem proves the result for a continuous linear operator \(S\) on a Banach space \(X\) instead of \(T_\nu\). This theorem is closely related to recent work by Jarosz and Villena about the characterization of topologies on function algebras and function spaces which make a fixed operator continuous. The last section presents a further extension for multipliers.