Operators preserving the representation of a semidirect product group

Abstract

For a locally compact abelian group $\text{R}$ with a uniform lattice $\Lambda$ and a group $G$ that acts on $\text{R}$ by continuous automorphisms, we study operators commuting with the representation of $G⋊\Lambda$ on $L^2(\text{R})$. As a consequence, we give a characterization of shift-dilation preserving operators.