 ### some small refinements

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 ... ... @@ -222,7 +222,7 @@ In other words $\benbrace*{D,g}$ is acting via the following element of $C_c^\in The computation in the proof of \cref{lem:symbol_welldefined} now shows: \begin{lemma} If$D$is an operator of order$1$, the symbol can be computed as If$D$is an operator of order$1$, the symbol can be computed as\marginnote{again, depending on notation the$i$-term might not come up} $\sigma_1(D)(\mathd g) u = i \,\benbrace*{D,g} u$ ... ... @@ -230,7 +230,7 @@ The computation in the proof of \cref{lem:symbol_welldefined} now shows: The symbol gives raise to an extremly important class of differential operators: \begin{definition} \begin{definition}\label{def:elliptic} Let$D \in \DiffOp^k(E_0,E_1)$and$x \in M$. Then we say, that$D$is \Index{elliptic at}$x$if for each$\xi \in \Tang_x^* M$,$\xi \neq 0$the homomorphism$\sigma_k(D)(\xi) \colon (E_0)_x \to (E_1)_x$is invertible.$D$is \Index{elliptic}, if it is elliptic at each point$x \in M$. ... ... @@ -267,13 +267,13 @@ Before applying the techniques of \cref{sec:unbounded_operators} to them, we fir for$u$and$v$compactly supported. \end{definition} \begin{theorem} Each differential operator$D \in \DiffOp^k(E_0,E_1)$has a formal adjoint in$D^\dagger \in \DiffOp^k(E_1,E_0)$and the formal adjoint is uniquely defined. \begin{theorem}\label{thm:existence_formal_adjoint} Each operator$D \in \DiffOp^k(E_0,E_1)$has a formal adjoint$D^\dagger \in \DiffOp^k(E_1,E_0)$and the formal adjoint is uniquely defined. Furthermore the symbol of the formal adjoint can be computed pointwise:\marginnote{depending on conventions regarding the symbol, there might be a minus sign here, cf.\ \cite[Prop.~10.1.4]{higson2000analytic}} $\sigma_k(D^\dagger)(\xi) = \enbrace[\big]{\sigma_k(D)(\xi)}^*$ Here the adjoint is meant as the vector bundle homomorphisms adjoint, which exists since our bundles are equipped with Hermitian bundle metrics. Here the adjoint is meant as the vector bundle homomorphism adjoint, which exists since our bundles are equipped with Hermitian bundle metrics. \end{theorem} \begin{proof} See \cite[Thm.~2.3.6]{ebert_index_lec}. ... ... @@ -562,7 +562,7 @@ In particular, it follows that the eigenvalues for a discrete subset of$\mathbb \begin{proof} The first two statements are proven by standard arguments as for selfadjoint endomorphisms of finite-dimensional Hilbert spaces. The third statement deepends on elliptic regularity: The third statement depends on elliptic regularity: Let $x \in U_\Lambda$. Since $D-\lambda$ is elliptic,\todo{why?} all eigenfunctions an hence $x$ are smooth. We write $x$ as an orthogonal sum $x = \sum_{\abs*{\lambda} \le \Lambda} x_\lambda$ and compute ... ...
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