Abstract. The paper concerns algebras of almost periodic pseudodifferential operators on R d with symbols in Hörmander 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 L 2 and on the Besicovitch Hilbert space of almost periodic functions.
IntroductionThe paper is a study of three representations of almost periodic (a.p.) pseudodifferential operators on R d with applications in the spectral theory of such operators. The symbols a ∈ C ∞ (R 2d ) we study belong to Hörmander classes S m ρ,δ with 0 < ρ ≤ 1, 0 ≤ δ < 1, δ ≤ ρ and m ∈ R. We are interested in the case when a(·, ξ) is a continuous a.p. function in the sense of Bohr for all ξ ∈ R d , and study the corresponding algebra of pseudodifferential operators in the Kohn-Nirenberg quantization, denoted AP L ∞ ρ,δ . Ruzhansky and Turunen [23] have studied the related class of pseudodifferential operators on the torus T d where the almost periodicity is replaced by periodicity. Pseudodifferential operators on more general compact Lie groups are studied in their monograph [24].In an earlier paper [31] we have shown that the transformation of a symbol a, defined by U(a)(ξ) λ,λ ′ = M x (a(x, ξ − λ ′ )e −2πix·(λ ′ −λ) ), λ, λ ′ , ξ ∈ R d