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