# Relative compactness of differential operator

Let $$\Omega$$ be $$\mathbb R^n$$ or a complete (unbounded) open manifold, and $$f$$ be a smooth function on $$\Omega$$.

We consider a self-adjoint 2nd. elliptic operator $$H$$ on $$L^2$$ space(to simplify the case, one can regard $$H$$ as Lapalacian).

We assume that $$f\to 0,~df \to0$$ as $$x\to \infty$$.

Q Is this enough to show that $$f\cdot\nabla$$ is $$H$$-compact, i.e. $$f\nabla(H+i\lambda)^{-1}$$ is a compact operator, for some $$\lambda\in\mathbb R_+$$.

PS: I am not sure the problem was studied or not. It will also be greatly helpful, if anyone give some reference.