Let $T$ be the operator from $L^2(\mathbb R^n)$ to $L^2(\mathbb R^n)$
defined as composition of convolution and multiplication, $Tf := (af) * g$ where $g$ is in $L^2$ and $a$ is a bounded function.
Can we find the spectrum of $T$? For $a$ identically equal 1, the spectrum is the essential range of the Fourier transform of $g$. I am interested in the more general case. If both $a$ and the Fourier transform of $g$ are positive functions, I assume that the spectrum of $T$ will also be positive but don't have a proof.