Convolutions for localization operators

Abstract

Quantum harmonic analysis on phase space is shown to be linked with localization operators. The convolution between operators and the convolution between a function and an operator provide a conceptual framework for the theory of localization operators which is complemented by an appropriate Fourier transform, the Fourier-Wigner transform. We use Lieb's uncertainty principle to establish a sharp Hausdorff-Young inequality for the Fourier-Wigner transform. Noncommutative Tauberian theorems due to Werner allow us to extend results of Bayer and Gr\"ochenig on localization operators. Furthermore we show that the Arveson spectrum and the theory of Banach modules provide the abstract setting of quantum harmonic analysis.