Let $L$ be a positive self-adjoint operator defined densely on $L^2(M)$ where $M$ is a compact manifold. Also, let $\mathcal{D}(L) \subset H^1(M)$. It is known that $\mathcal{D}(L) \subset \mathcal{D}((L)^{1/2})$ as a continuous inclusion. I am trying to see whether this inclusion is also compact. I suspect that this has something to do with Rellich's theorem and interpolation spaces, but cannot work out a rigorous proof. Any help would be appreciated.

Edit: I can also work out that $L$ has compact resolvent.

Further edit: Please take a look at my proposed answer below.