Band-dominated and Fourier-band-dominated operators on locally compact abelian groups

Abstract

Abstract By relating notions from quantum harmonic analysis and band-dominated operator theory, we prove that, over any locally compact abelian group 𝐺, the operator algebra C 1 \mathcal{C}_{1} from quantum harmonic analysis agrees with the intersection of band-dominated operators and Fourier band-dominated operators. As an application, we characterize the compactness of operators acting on L 2 ⁢ ( G ) L^{2}(G) as well as the modulation spaces M p ⁢ ( G ) M^{p}(G) and compare our findings with previous results in the discrete case. In particular, our results can be interpreted as a generalization of the limit operator concept to the non-discrete world. Moreover, we briefly discuss property A ′ A^{\prime} for arbitrary locally compact abelian groups.