 ### some work towards complete diff ops

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 ... ... @@ -149,7 +149,7 @@ There is a coordinate-free description as well: $\benbrace*{\mathd,f} \omega = \mathd(f \omega) - f(\mathd \omega) = (\mathd f) \wedge \omega + f \mathd \omega - f \mathd \omega = (\mathd f) \wedge \omega$ Again via \cref{thm:diff_op_coordfree} we see that $\mathd$ is a differential operator of order one. Again via \cref{thm:diff_op_coordfree} we see that $\mathd$ is a differential operator of order 1. \item Let $E \to M$ be an arbitrary vector bundle with a \Index{connection} $\nabla \colon \Gamma(M,E) \to \Gamma(M, \Tang^*M \otimes E)$, i.e. $\nabla$ is linear and fulfils $\nabla(f s)= \mathd f \otimes s + f \nabla s$ for $f \in C^\infty(M)$ and $s \in \Gamma(M,E)$. In particular $... ... @@ -298,7 +298,7 @@ Furthermore a symmetric differential operator may be \Index{selfadjoint}, i.e.\ 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: \begin{example} \begin{example}\label{ex:max_domain_neq_min_domain} Consider the open 1-manifold M=(0,1) (with the usual Riemannian metric) and D= i \diffd{}{x} (we need the factor i here for D to be symmetric, use integration by parts to see this). The Hilbert space we get is L^2[0,1]. ... ... @@ -310,15 +310,17 @@ As the following example shows, all three operators may be different: 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: \begin{lemma} \begin{lemma}\label{lem:symmetric_ess_sa} Let D be a symmetric differential operator and u \in L^2(M,E) \emph{compactly supported}. Then u belongs to the minimal domain of D if and only if it belongs to the maximal domain. \end{lemma} In particular we see, that every symmetric differential operator on a closed manifold is essentially selfadjoint. More generally every compactly supported, symmetric differential operator on an open manifold is essentially selfadjoint. \begin{corollary} Every symmetric differential operator on a closed manifold is essentially selfadjoint. More generally every compactly supported, symmetric differential operator on an open manifold is essentially selfadjoint. \end{corollary} \begin{proof} \begin{proof}[Proof of \cref{lem:symmetric_ess_sa}] Let K be a compact subset of M. We need \emph{Friedrichs' mollifiers} to proceed: A local variant is discussed in \cref{prop:friedrichs}, which can be globalised via the usual arguments to operators F_t \colon L^2(K,E) \to L^2(M,E) such that ... ... @@ -333,11 +335,20 @@ More generally every compactly supported, symmetric differential operator on an \[ D F_t u = F_t D u + \benbrace*{D,F_t} u$ By \cref{rmk:max_domain_distribution} and (a) we know, that the first summand is uniformly bounded ; the second one is uniformly bounded by (d). By \cref{rmk:max_domain_distribution} and (a) we know, that the first summand is uniformly bounded; the second one is uniformly bounded by (d). Considering $t=\frac{1}{n}$ we get a sequence of elements $u_n$ in the domain of $D$ converging to $u$ such that $\norm*{D u_n}$ is uniformly bounded. Hence \cref{lem:charac_minimal_domain} implies, that $u$ is in the minimal domain of $D$. \end{proof} As seen in \cref{ex:max_domain_neq_min_domain} the selfadjointness of differential operators on open manifolds is a more delicate matter. We need to look at multiplication operators. \begin{definition} Let $D$ be a differential operator on $M$. 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. ... ... @@ -345,8 +356,6 @@ More generally every compactly supported, symmetric differential operator on an \begin{proof} \todo{add proof} \end{proof} \todo[inline]{include wave operator stuff as well?} % section hilbert_space_techniques_and_the_formal_adjoint (end) \section{Analysis of Elliptic Differential Operators} % (fold) ... ...
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