t the range $R(I+ P_0)$ is dense that is, \[evo17-a\] =L\^2(GSI).

As in the proof of Lemma \[csdale0\] (note that here the assumptions are somewhat weaker), we have for all $\phi\in D(P_0)$, (we write $L^2=L^2(G\times S\times I)$) \[ineq\] & P\_0,\_[L\^2]{} =-[E]{},\_[L\^2]{} +\_x,\_[L\^2]{}+ CS\_0,\_[L\^2]{}\
=& (-+CS\_0),\_[L\^2]{} + ,\_[T\^2(\_+)]{}\^2 +[12]{}\_[GS]{}S\_0(,0)(,,0)\^2 dx d, which in combination with the assumptions , implies $${\left\langle}P_0\phi,\phi{\right\rangle}_{L^2(G\times S\times I)}\geq 0,\quad \forall \phi\in D(P_0).$$

If $\phi\in D(\tilde{P}_0)$, choose a sequence $\phi_n\in D(P_0)$ such that $\phi_n\to \phi$ and $P_0\phi_n\to \tilde{P}_0\phi$ in $L^2(G\times S\times I)$ when $n\to\infty$. By the above inequality, we have $${\left\langle}\tilde{P}_0\phi,\phi{\right\rangle}_{L^2(G\times S\times I)}
=\lim_n {\left\langle}P_0\phi_n,\phi_n{\right\rangle}_{L^2(G\times S\times I)}\geq 0,$$ which gives .

Finally, from it follows that \[evo18-ab\] [(I+P\_0)]{}\_[L\^2(GSI)]{}\_[L\^2(GSI)]{},D(P\_0). and therefore $R(I+\tilde P_0)$ is closed in $L^2(G\times S\times I)$. This result, the observation that $R(I+P_0)\subset R(I+\tilde{P}_0)$, and show that $R(I+\tilde P_0)=L^2(G\times S\times I)$. The proof is complete.

Theorem \[md-evoth\] says that:

\[d-cor\] The operator $-\tilde P_0:L^2(G\times S\times I)\to L^2(G\times S\times I)$ is $m$-dissipative, or equivalently $\tilde P_0$ is $m$-accretive.

See e.g. [@dautraylionsv5 p. 340].

\[F-L-P-R\] The general theory of initial boundary value problems of symmetric formally dissipative first order partial differential operators can alternatively be applied to show the $m$-dissipativity of $-\tilde P_0$. The (classical) results for positive symmetric initial boundary value problems can be found in [@lax], Theorem 3.2 and discussion in section 4 therein; [@friedrich58], together with discussion in section 17 therein; [@sarason] and [@rauch85]. The spatial domain there is replaced with $G\times S$ (here the additional smooth compact manifold $S