Selfadjointness of the Dirac operator

If $T$ is a densely defined operator on a Hilbert space $\sH$, its adjoint $T^*$ has domain

$$\Dom T^* := \set{\phi \in \sH : \exists \chi \in \sH \text... ...{\phi} = \braket{\psi}{\chi} \text{ for all } \psi \in \Dom T}$$

and then $T^*\phi := \chi$, of course, so that the formula $\braket{T\psi}{\phi} = \braket{\psi}{T^*\phi}$ holds. If $T$ is symmetric, then clearly $\Dom T \subseteq \Dom T^*$ with $T^* = T$ on $\Dom T$: that is, $T^*$ is an extension of $T$ to a larger domain.

The second adjoint $T^{**} =: \Bar{T}$ is called the closure of $T$ (symmetric operators always have this closure), where the domain of the closure is
$$\begin{align*} \Dom\Bar{T} := \set{\psi \in \sH : \exists \phi\in \sH &\text{ a... ...at } \psi_n \to \psi \text{ and } T\psi_n \to \phi \text{ in } \sH} \end{align*}$$
In other words, the graph of $\Bar{T}$ in $\sH \oplus \sH$ is the closure of the graph of $T$. And then, of course, we put $\Bar{T}\psi := \phi$. When $T$ is symmetric, we get

$$\Dom T \subseteq \Dom\Bar{T} \subseteq \Dom T^*.$$

The main result of this chapter is that the Dirac operator on a compact Riemannian spin manifold is essentially selfadjoint. This was proved by Wolf in 1973; he actually showed the result also for noncompact manifolds which are complete with respect to the Riemannian distance given by the metric [Wolf]. In his proof, completeness is needed to establish that closed geodesic balls are compact; that proof is also given in the book by Friedrich [Fri]. For simplicity, we deal here only with the compact case.