Commit 62955af5 authored by Jannes Bantje's avatar Jannes Bantje

harmonised spelling of „selfadjoint“

parent c9c1c228
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......@@ -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 $D$.
......@@ -523,7 +523,7 @@ To finally prove the Fredholm property we need the following result from functio
\todo[inline]{add Hodge decomposition if needed}
\begin{theorem}{Spectral Decomposition}\label{thm:elliptic_spectral}
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 $k \ge 1$.
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 $k \ge 1$.
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)]
......@@ -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 $\norm*{x}_0 \le 1$ 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:
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