How to find spectrum of a convolution operator

Question

Say $k$ be s.t. $\hat{k}$ is a bounded function on an LCA group $G$ and $Tf=f*k$. Then $T$ is bounded on $L^2(G)$. Is there anything I can say about $\sigma(T)$? (except the properties that follow from $T$ being bounded)

Answer 1

Actually convolution with $k$ defines a bounded operator on $L^2(G)$ if and only if $\hat k$ is bounded. In this case the spectrum is exactly the essential range of $\hat k$.

In fact if $m$ is any bounded measurable function on $\hat G$ then $m$ defines a bounded operator on $L^2(G)$ by $$\widehat{Tf}=m\hat f,$$whether or not there exists $k$ with $m=\hat k$. And the spectrum is the essential range of $m$; this is clear from the definitions if you note that $1/(m-\lambda)$ is essentially bounded if and only if $\lambda$ is not in the essential range of $m$.