Representations of almost periodic pseudodifferential operators and applications in spectral theory

TL;DR

This paper investigates the structure and representations of almost periodic pseudodifferential operators with H"ormander class symbols, establishing unitary equivalences and applying findings to spectral theory in various function spaces.

Contribution

It introduces and compares three representations of these operators, proving unitary equivalence for nonpositive order, and applies the results to spectral analysis.

Findings

Two representations are unitarily equivalent for nonpositive order

The paper extends spectral theory to operators on $L^2$ and Besicovitch spaces

Provides new tools for analyzing almost periodic pseudodifferential operators

Abstract

The paper concerns algebras of almost periodic pseudodifferential operators on $\mathbb R^d$ with symbols in H\"ormander 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.