Representations of almost periodic pseudodifferential operators and applications in spectral theory

Abstract

The paper concerns algebras of almost periodic pseudodifferential operators on \({\mathbb{R}^d}\) with symbols in Hormander classes. We study three representations of such algebras, one of which was introduced by Coburn, Moyer and Singer and the other two inspired by results in probability theory by Gladyshev. Two of the representations are shown to be unitarily equivalent for nonpositive order. We apply the results to spectral theory for almost periodic pseudodifferential operators acting on L2 and on the Besicovitch Hilbert space of almost periodic functions.