Measure-operator convolutions and applications to mixed-state Gabor multipliers

TL;DR

This paper extends the framework of quantum harmonic analysis to include measure-operator convolutions, enabling new insights into eigenvalue distributions, inequalities, and approximation properties of Gabor multipliers.

Contribution

It introduces a generalized convolution framework for measures and operators, leading to novel results in eigenvalue distribution, inequalities, and approximation of localization operators.

Findings

Eigenvalue distribution results for mixed-state Gabor multipliers

A version of the Berezin-Lieb inequality for lattices

Continuity and approximation properties of Gabor multipliers

Abstract

For the Weyl-Heisenberg group, convolutions between functions and operators were defined by Werner as a part of a framework called quantum harmonic analysis. We show how recent results by Feichtinger can be used to extend this definition to include convolutions between measures and operators. Many properties of function-operator convolutions carry over to this setting and allow us to prove novel results on the distribution of eigenvalues of mixed-state Gabor multipliers and derive a version of the Berezin-Lieb inequality for lattices. New results on the continuity of Gabor multipliers with respect to lattice parameters, masks and windows as well as their ability to approximate localization operators are also derived using this framework.