Integral representation of translation-invariant operators on reproducing kernel Hilbert spaces
We suppose that $G$ is a locally compact abelian group, $Y$ is a measure space, and $H$ is a reproducing kernel Hilbert space on $G\times Y$ such that $H$ is naturally embedded into $L^2(G\times Y)$ and it is invariant under the translations associated with $G$. We consider the von Neumann algebra of all bounded linear operators acting on $H$ that commute with these translations. Assuming that this algebra is commutative, we represent its elements as integral operators and characterize the corresponding integral kernels. Furthermore, we give W*-algebra structure on the functions associated with the integral kernels. We apply this general scheme to a series of examples, including rotation- or translation-invariant operators in Bergman or Fock spaces.