Usually, Spectral Theory for Operators in Hilbert Spaces is defined over $\mathbb{C}$, the complex field. I'm trying to study some results of Spectral Theory in the real field. Specifically, I have the Laplace-Beltrami operator, $\Delta_g$, defined on $L^2(S^*M)$, where $S^*M$ is the unit cotangent bundle of a Riemannian manifold. It is clear that $L^2(S^*M)$ is a real Hilbert Space (as functions over a Riemannian manifold are real-valued), so I can't apply usual statements of Compact Operators (as they are stated in the context of complex Hilbert spaces). My main goal is to say that the spectrum of $\Delta_g$ is discrete, and that an orthonormal basis of eigenfunctions of $\Delta_g$ exists.
I have considered two main approaches:
Consider the complexification of $L^2(S^*M)$ and apply the usual theorem for complex Hilbert Spaces.
Defining an involution over $L^2(S^*M)$ as a real Hilbert space and apply that $\Delta_g$ is a symmetric (self-adjoint) operator.
I am uncertain whether there are references that address this kind of results in a more general setting beyond the specific case of $\Delta_g$. Any help would be appreciated.