Let $T_g\in B(L^2(\mathbb{R}))$ be defined by $T_g(f):=g*f$ for $f\in L^2(\mathbb{R})$, where $g\in L^2(\mathbb{R})$ is given such that $||g||_2\neq 0$. I am aware of the trick to analyzing the spectrum of $T$ by using the fact that $B:=\mathcal{F}T_g\mathcal{F}^{-1}$ has the same spectrum as $T_g$ where $\mathcal{F}$ is the Fourier transform on $L^2(\mathbb{R})$, and that $B$ is a multiplication operator: $B(f)=\hat{g}(\xi)f(\xi)$; from this one then concludes that the spectrum of $T_g$ is the essential range of $\hat{g}$.
In general, $\hat{g}$ will not be constant, and therefore I conclude from this that the point spectrum of $T_g$ is empty. Is this assertion correct or am I missing something? More generally, what can be said about the continuous spectrum and the residual spectrum of $T_g$?