@@ -28,7 +28,7 @@ Its domain is called the \Index{minimal domain} of $T$ and has the following pro

Then $u \in\Hilb$ belongs to the domain of $\overline{T}$ if and only if there is a sequence $\set*{u_j}_{j=1}^\infty\subset\domain(T)$ such that $u_j \to u$ while $\norm*{T u_j}$ remains bounded.

\end{lemma}

\begin{proof}

See \cite[Lem.~1.8.1]{higson2000analytic}.

See \cite[Lem.~1.8.1]{higson_roe}.

\end{proof}

Usually the overline to denote the closure of $T$ is ommitted (this seldomly causes confusion).

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@@ -75,7 +75,7 @@ The interest in selfadjoint operators mostly originates from the following prope

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}}

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]{higson_roe}}

@@ -389,8 +389,8 @@ In particular this will allow us to view differential operators as Fredholm oper

\begin{remark}

In \cref{sec:convolution_and_the_fourier_transform} the Fourier transform plays a central role.

Some authors (e.g.\ \textcite{higson2000analytic}) prefer the simpler Fourier \emph{series} and therefore establish the theory on the torus $\mathbb{T}^n$ first, which has the discrete group $\mathbb{Z}^n$ as Pontrjagin dual.

These results can be \enquote{grafted} onto a general manifold $M$ as well, although \textcite{higson2000analytic} do not specify how this is done.

Some authors (e.g.\ \textcite{higson_roe}) prefer the simpler Fourier \emph{series} and therefore establish the theory on the torus $\mathbb{T}^n$ first, which has the discrete group $\mathbb{Z}^n$ as Pontrjagin dual.

These results can be \enquote{grafted} onto a general manifold $M$ as well, although \textcite{higson_roe} do not specify how this is done.

\end{remark}

Firstly we need to define Sobolev spaces on manifolds.

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@@ -577,7 +577,7 @@ In particular, it follows that the eigenvalues for a discrete subset of $\mathbb

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

These notes follow the book \citetitle{higson2000analytic} by \textcite{higson2000analytic} and will also try to cover most of the lecture notes on index theory by Johannes Ebert \cite{ebert_index_lec}.

These notes follow the book \citetitle{higson_roe} by \textcite{higson_roe} and will also try to cover most of the lecture notes on index theory by Johannes Ebert \cite{ebert_index_lec}.

They should also cover the most important aspects of \citetitle{lawson_spin} by \textcite{lawson_spin}.

\todo[inline]{what are other suitable references?}