Commit dbc880f6 authored by Jannes Bantje's avatar Jannes Bantje

rename bib key

parent 8fd3de46
Pipeline #52279 canceled with stages
%!TEX root = ../index_theory.tex
\chapter{\K-Homology of an Elliptic Differential Operator} % (fold)
\label{cha:_k_homology_of_an_elliptic_differential_operator}
\todo[inline]{should incorporate \cite[10.6+]{higson2000analytic} plus more background if needed}
\todo[inline]{should incorporate \cite[10.6+]{higson_roe} plus more background if needed}
% chapter _k_homology_of_an_elliptic_differential_operator (end)
\ No newline at end of file
......@@ -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).
......@@ -75,7 +75,7 @@ The interest in selfadjoint operators mostly originates from the following prope
The spectrum of a selfadjoint operator is real.
\end{lemma}
\begin{proof}
See \cite[Lem.~1.8.4]{higson2000analytic}.
See \cite[Lem.~1.8.4]{higson_roe}.
\end{proof}
\todo[inline]{mention Cayley transform, functional calculus?}
......@@ -269,7 +269,7 @@ Before applying the techniques of \cref{sec:unbounded_operators} to them, we fir
\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}}
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}}
\[
\sigma_k(D^\dagger)(\xi) = \enbrace[\big]{\sigma_k(D)(\xi)}^*
\]
......@@ -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.
......@@ -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:
\begin{proposition}{\cite[Prop.~10.4.5]{higson2000analytic}}
\begin{proposition}{\cite[Prop.~10.4.5]{higson_roe}}
Let $D$ be as in \cref{thm:elliptic_spectral}.
If $\varphi \in C_0(\mathbb{R})$ the operator $\varphi(D) \colon L^2(M,E) \to L^2(M,E)$ is compact.
\end{proposition}
......
......@@ -16,7 +16,7 @@
% \input{contents/declaration.tex}
% \input{contents/preface.tex}
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?}
......
......@@ -570,7 +570,7 @@
eprinttype={arxiv},
eprint={math/0701484v4},
}
@book{higson2000analytic,
@book{higson_roe,
title={Analytic \K-homology},
author={Higson, Nigel and Roe, John},
year={2000},
......
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