Reduction of operators with almost periodic symbols to operators overC *-Algebras on sections of associated bundles over a torus

Abstract

A decomposition of any pseudodifferential operator o(D) on R with almost periodic symbol as a(D) = oA(D) ® 1 is obtained in the paper, where oA(D) is a pseudodifferential operator over a certain C*-algebra A acting on sections of a vector bundle over a torus T whose fibre is A. The coincidence of spectra sp (D) = sp UA(D) is proved for all a(D) either bounded or elliptic.

Almost periodic pseudodifferential operators and Gevrey classes

“…pseudodifferential operators (ΨDO) on R d are operators defined by Hörmander type symbols S m ρ,δ that are almost periodic in the space variables. They have been studied thoroughly, in particular by M. A. Shubin [32][33][34][35][36], and by Coburn, Moyer and Singer [3], Dedik [5], Filippov [7], Pankov [22], Rabinovich [23] and Wahlberg [39]. The related class of pseudodifferential operators on the torus T d (where the almost periodicity is replaced by periodicity) is treated by Ruzhansky and Turunen [28] (see also their recent monograph [29]).…”

Representations of almost periodic pseudodifferential operators and applications in spectral theory

“…Such operators are called a.p. pseudodifferential operators, and have been investigated by Coburn, Moyer and Singer [3], Dedik [5], Filippov [8], Oliaro, Rodino and Wahlberg [17], Pankov [18], Rabinovich [19], Shubin [25][26][27][28][29], and others. Recently, 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.…”