# Discrete convolution operators and Riesz systems generated by actions of abelian groups

## TL;DR

This paper characterizes bounded translation-commuting endomorphisms on vector-valued -spaces over discrete abelian groups, linking them to matrix-valued functions on the dual group, and explores their invertibility and norm properties.

## Contribution

It provides a complete description of the algebra of such endomorphisms and characterizes their invertibility, extending the understanding of Riesz systems generated by abelian group actions.

## Key findings

- Endomorphisms form a C*-algebra isomorphic to matrix-valued essentially bounded functions.
- Criteria for invertibility of these endomorphisms are established.
- Explicit formulas for norms and inverse norms are derived.

## Abstract

We study the bounded endomorphisms of $\ell_{N}^2(G)=\ell^2(G)\times \dots \times\ell^2(G)$ that commute with translations, where $G$ is a discrete abelian group. It is shown that they form a C*-algebra isomorphic to the C*-algebra of $N\times N$ matrices with entries in $L^\infty(\widehat{G})$, where $\widehat{G}$ is the dual space of $G$. Characterizations of when these endomorphisms are invertible, and expressions for their norms and for the norms of their inverses, are given. These results allow us to study Riesz systems that arise from the action of $ G $ on a finite set of elements of a Hilbert space.