@@ -283,7 +283,7 @@ The closure is written $\overline{D}$ as usual, although the overline is sometim

Recall that the domain of $\overline{D}$ is called the \Index{minimal domain}.

In these notes we will be primarily interested in \Index{symmetric} operators, which in the case of differential operators just means that $D=D^\dagger$, compare \cref{def:symm_selfad}.

In the context of differential operators this property is also called \Index{formally self-adjoint}.

In the context of differential operators this property is also called \Index{formally selfadjoint}.

Being symmetric allows us to exhibit the \Index{adjoint}$D^*$ as an extension of $\overline{D}$.

Recall that the domain of $D^*$ is called the \Index{maximal domain} of $D$.

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@@ -523,7 +523,7 @@ To finally prove the Fredholm property we need the following result from functio

Let $M$ be a closed Riemannian manifold, $E \to M$ a hermitian vector bundle and $D \colon\Gamma(M,E)\to\Gamma(M,E)$ be a formally self-adjoint elliptic differential operator of order $k \ge1$.

Let $M$ be a closed Riemannian manifold, $E \to M$ a hermitian vector bundle and $D \colon\Gamma(M,E)\to\Gamma(M,E)$ be a formally selfadjoint elliptic differential operator of order $k \ge1$.

For $\lambda\in\mathbb{C}$ let $V_\lambda\coloneqq\ker(D -\lambda)\subset L^2(M,E)$ be the eigenspace of $D$ to the eigenvalue $\lambda$.

Then the following statements hold

\begin{enumerate}[(i)]

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@@ -548,7 +548,7 @@ In particular, it follows that the eigenvalues for a discrete subset of $\mathbb

From this we see, that the set of all $x$ with $\norm*{x}_0\le1$ is bounded with respect to $\norm*{\cdot}_k$.

Therefore Rellich compactness implies that the $\norm*{\cdot}_0$-unit ball in $U_\Lambda$ is relatively compact, so that $U_\Lambda$ is finite dimensional.

See \cite[52]{ebert_index_lec} for a proof of the last statement (this requires the spectral theorem for self-adjoint compact operators).

See \cite[52]{ebert_index_lec} for a proof of the last statement (this requires the spectral theorem for selfadjoint compact operators).

\end{proof}

Using functional calculus \cref{thm:elliptic_spectral} easily gives the following: