EDIT: This solution does not satisfy the third condition, which rules out the Hilbert transform itself. So, this is an answer to different question. I do not delete it in hope it may be useful for someone.
Let $\phi(x)$ be a smooth monotone function such that $x-\phi(x)$
has compact support. This is a diffeomorphism of the real line
and the pullback $\phi^*$ is a linear operator acting on
$L^2(\mathbb R)$. The singular integral operator
$$(\phi^*)^{-1}{\mathcal H}\phi^*$$
has all the properties you need.
