On Convolution Dominated Operators

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 $f\in L^2(G)$ $|Af(x)|\le a\ast |f|(x)$, 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 $C{D}_{reg}(G)$. For amenable $G$ which is rigidly symmetric as a discrete group, we show that any element of $C{D}_{reg}(G)$ is invertible in $C{D}_{reg}(G)$ 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 $L^2(E)$.

22pages, to appear in Integral Equations and Operator Theory