@@ -313,7 +313,7 @@ Using distribution theory\footnote{see \url{https://en.wikipedia.org/wiki/Distri

To make sense of the expression $D u$ here, one has to use distribution theory, i.e. the \Index{distribution}$D u$ is required to be an actual Element of $L^2(M,E)$.

\end{remark}

Furthermore a symmetric differential operator may be \Index{selfadjoint}, i.e.\ fullfil $\overline{D}=D^*$.

Furthermore a symmetric differential operator may be \Index{essentially selfadjoint} (or just \Index{selfadjoint}), i.e.\ fullfil $\overline{D}=D^*$.

Note that the closure $\overline{D}$ is an extension of $D$ in general and that $D^*$ is an extension of $\overline{D}$ in the symmetric case.

As the following example shows, all three operators may be different:

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@@ -327,7 +327,18 @@ As the following example shows, all three operators may be different:

\end{example}

This example already suggests, that non-selfadjointness is in some way connected to \enquote{boundary conditions} on a non-compact manifold $M$.

The same phenomenon shows up in the following statement:

We investigate this phenomenon in the next section and conclude the present section with the following statement about elliptic operators.

\begin{lemma}\label{lem:adjoint_elliptic}

If $D$ is a symmetric, essentially selfadjoint elliptic operator, then its adjoint is elliptic as well.

Let $D$ be a symmetric differential operator and $u \in L^2(M,E)$\emph{compactly supported}.

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@@ -367,15 +378,9 @@ We need to look at multiplication operators.

Then $M$ is called \Index{complete} for $D$ if there is a smooth proper function $g \colon M \to\mathbb{R}$ such that $\benbrace*{D,\rho(g)}$ is a bounded Hilbert space operator.

\end{definition}

\todo[inline]{include wave operator stuff as well?}

\begin{lemma}\label{lem:adjoint_elliptic}

If $D$ is a symmetric elliptic operator, then its adjoint is elliptic as well.