Translation preserving operators on locally compact abelian groups

TL;DR

This paper investigates translation preserving operators on locally compact abelian groups, establishing a correspondence with range operators and providing conditions for these operators to be Hilbert-Schmidt or finite trace.

Contribution

It introduces a novel framework linking translation preserving operators with range operators and characterizes their Hilbert-Schmidt and trace properties.

Findings

One-to-one correspondence between translation preserving and range operators

Necessary conditions for Hilbert-Schmidt property

Criteria for finite trace operators

Abstract

We study translation preserving operators, that is operators commuting with translations by a closed subgroup of a locally compact abelian group. We show that there is a one to one correspondence between these operators and range operators. Furthermore, we obtain a necessary condition for a translation preserving operator to be Hilbert Schmidt or of finite trace in terms of its range operator.