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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.