On Modulation and Translation Invariant Operators and the Heisenberg Module

TL;DR

This paper explores operators invariant under phase space translations or modulations, linking them to Heisenberg modules, and characterizes their structure and composition through finite-rank limits and quantisation schemes.

Contribution

It provides a new characterization of translation and modulation invariant operators via limits of finite-rank operators within the framework of Heisenberg modules.

Findings

Operators can be approximated by finite-rank operators

Discrete representations facilitate understanding of operator composition

Different quantisation schemes are analyzed in relation to these operators

Abstract

We investigate spaces of operators which are invariant under translations or modulations by lattices in phase space. The natural connection to the Heisenberg module is considered, giving results on the characterisation of such operators as limits of finite--rank operators. Discrete representations of these operators in terms of elementary objects and the composition calculus are given. Different quantisation schemes are discussed with respect to the results.