Let $A$ be a densely defined unbounded self-adjoint operator defined on $L^2(\mathbb{T}^2)$, where $\mathbb{T}^2$ stands for the 2-torus. It is known that $A$ is positive, that is, $\langle Au, u\rangle \geq 0, u \in L^2(\mathbb{T}^2)$. Let $A^{1/2}$ denote the square root of $A$, defined by the spectral theorem, which says that apriori, $A^{1/2}$ is a densely defined self-adjoint operator on $L^2(\mathbb{T}^2)$. But can we say something more, like, $\mathcal{D}(A^{1/2}) \subset H^{1/2}(\mathbb{T}^2)$?

Basically, I am trying to see how this generalizes to generic (at least compact) Riemannian manifolds, and trying to understand a relatively simple case first. Any reference would be appreciated. Thanks!