Is Spectral Theory necessary to show that QM operators are densely defined, essentially self-adjoined?

Question

I'm reading Hall's Quantum Theory for Mathematicians, and I merely wish to check my understanding on a small thing:

Spectral Theory is needed to show that QM operators (e.g. the position or momentum operators) are defined on dense subsets of the Hilbert space $L^2(\mathbb{R})$ and are (essentially) self-adjoint.

Right?