Multiplication-invariant operators and the classification of LCA group frames

TL;DR

This paper characterizes multiplication-invariant operators on vector-valued L2 spaces, classifies frames generated by these operators, and applies the results to abelian group frames and translation-invariant operators.

Contribution

It provides a novel characterization of MI operators via range functions and classifies frames generated by multiplications, extending to group and translation-invariant contexts.

Findings

Characterization of MI operators through measurable range functions

Classification of frames of multiplications via unitary equivalence

Applications to abelian group frames and TI operators

Abstract

In this paper we study the properties of multiplication invariant (MI) operators acting on subspaces of the vector-valued space $L^2(X;\mathcal H)$. We characterize such operators in terms of range functions by showing that there is an isomorphism between the category of MI spaces (with MI operators as morphisms) and the category of measurable range functions whose morphisms are measurable range operators. We investigate how global properties of an MI operator are reflected by local pointwise properties of its corresponding range operator. We also establish several results about frames generated by multiplications in $L^2(X;\mathcal H)$. This includes the classification of frames of multiplications with respect to unitary equivalence by measurable fields of Gramians. Finally, we show applications of our results in the study of abelian group frames and translation-invariant (TI) operators acting on subspaces of $L^2(\mathcal G)$, where $\mathcal G$ is a locally compact group.