 ### split into another section

parent 834eb326
Pipeline #8032 passed with stages
in 41 seconds
 ... ... @@ -313,7 +313,7 @@ Using distribution theory\footnote{see \url{https://en.wikipedia.org/wiki/Distri To make sense of the expression $D u$ here, one has to use distribution theory, i.e. the \Index{distribution} $D u$ is required to be an actual Element of $L^2(M,E)$. \end{remark} Furthermore a symmetric differential operator may be \Index{selfadjoint}, i.e.\ fullfil $\overline{D}=D^*$. Furthermore a symmetric differential operator may be \Index{essentially selfadjoint} (or just \Index{selfadjoint}), i.e.\ fullfil $\overline{D}=D^*$. Note that the closure $\overline{D}$ is an extension of $D$ in general and that $D^*$ is an extension of $\overline{D}$ in the symmetric case. As the following example shows, all three operators may be different: ... ... @@ -327,7 +327,18 @@ As the following example shows, all three operators may be different: \end{example} This example already suggests, that non-selfadjointness is in some way connected to \enquote{boundary conditions} on a non-compact manifold $M$. The same phenomenon shows up in the following statement: We investigate this phenomenon in the next section and conclude the present section with the following statement about elliptic operators. \begin{lemma}\label{lem:adjoint_elliptic} If $D$ is a symmetric, essentially selfadjoint elliptic operator, then its adjoint is elliptic as well. \end{lemma} \begin{proof} \todo{add proof} \end{proof} % section hilbert_space_techniques_and_the_formal_adjoint (end) \section{Selfadjointness of Differential Operators} % (fold) \label{sec:self_adjointness_of_differential_operators} \begin{lemma}\label{lem:symmetric_ess_sa} Let $D$ be a symmetric differential operator and $u \in L^2(M,E)$ \emph{compactly supported}. ... ... @@ -367,15 +378,9 @@ We need to look at multiplication operators. Then $M$ is called \Index{complete} for $D$ if there is a smooth proper function $g \colon M \to \mathbb{R}$ such that $\benbrace*{D,\rho(g)}$ is a bounded Hilbert space operator. \end{definition} \todo[inline]{include wave operator stuff as well?} \begin{lemma}\label{lem:adjoint_elliptic} If $D$ is a symmetric elliptic operator, then its adjoint is elliptic as well. \end{lemma} \begin{proof} \todo{add proof} \end{proof} % section hilbert_space_techniques_and_the_formal_adjoint (end) \todo[inline]{include wave operator stuff as well?} % section self_adjointness_of_differential_operators (end) \section{Analysis of Elliptic Differential Operators} % (fold) \label{sec:analysis_of_elliptic_differential_operators} ... ...
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!