Representations of almost periodic pseudodifferential operators and applications in spectral theory

Abstract

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

Nonharmonic Analysis of Boundary Value Problems

“…For further extensions of this periodic analysis to the almost periodic setting see e.g. Wahlberg [Wah09,Wah12]. The classical Fourier series on a circle T = R/Z can be viewed as a unitary transform in the Hilbert space L 2 (0, 1) generated by the operator of differentiation (−i d dx ) with periodic boundary conditions, because the system of exponents {exp(2πiλx), λ ∈ Z} is a system of its eigenfunctions.…”

Inverse-Closedness of Subalgebras of Integral Operators with Almost Periodic Kernels

“…This paper is devoted to almost periodic linear operators, i. e., operators with almost periodic 'coefficients' (we interpret the kernel of an integral operator as a kind of 'coefficients'). Different properties of such operators were investigated in [1,2,7,8,15,16,17,21,39,40,42,43,44,45,46,48,49,50,51,52,53,56] and other works. Equations and operators with almost periodic 'coefficients' often arise in applications for the following reason.…”