On Convolution Dominated Operators

TL;DR

This paper studies convolution dominated operators on $L^2(G)$ for locally compact groups, focusing on their algebraic properties, invertibility, and differences between discrete and non-discrete groups.

Contribution

It introduces the subalgebra of regular convolution dominated operators and proves invertibility results for amenable, rigidly symmetric groups.

Findings

Invertibility of $CD_{reg}(G)$ elements on $L^2(G)$ for certain groups

Existence of symmetric groups where convolution dominated operators are not inverse-closed

Analysis of algebraic structure of convolution dominated operators on different groups

Abstract

For a locally compact group $G$ we consider the algebra $CD(G)$ of convolution dominated operators on $L^{2}(G)$: An operator $A:L^2(G)\to L^2(G)$ is called convolution dominated if there exists $a\in L^1(G)$ such that for all $f \in L^2(G)$ $ |Af(x)| \leq a * |f| (x)$, for almost all $x \in G$. In the case of discrete groups those operators can be dealt with quite sufficiently if the group in question is rigidly symmetric. For non-discrete groups we investigate the subalgebra of regular convolution dominated operators $CD_{reg}(G)$. For amenable $G$ which is rigidly symmetric as a discrete group, we show that any element of $CD_{reg}(G)$ is invertible in $CD_{reg}(G)$ if it is invertible as a bounded operator on $L^2(G)$. We give an example of a symmetric group $E$ for which the convolution dominated operators are not inverse-closed in the bounded operators on $L^2(E)$.