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.