the independence of the parameter $\rho.$

\[hyp:independence\] Let $\rho_{0}>0$ and $\mathcal{M},\mathcal{N},\mathcal{M}'$ evolutionary at $\rho_{0}$ such that

(a) $\forall\varphi\in C_{c}^{\infty}({\mathbb{R}};H):\,\mathcal{M}\varphi'+\mathcal{M}'\varphi=(\mathcal{M}\varphi)'$ in the sense of distributions.

(b) There exists $c>0$ such that for all $\rho\geq\rho_{0}$ $${\operatorname{Re}}\langle(\partial_{t,\rho}\mathcal{M}+\mathcal{N})\varphi,\chi_{\mathbb{R}_{\leq a}}\varphi\rangle_{\rho}\geq c|\chi_{\mathbb{R}_{\leq a}}\varphi|_{\rho}^{2}$$ for each $\varphi\in C_{c}^{\infty}({\mathbb{R}};H)$ and each $a\in{\mathbb{R}}.$

Moreover, let $A\subseteq H\times H$ be maximal monotone with $(0,0)\in A.$

We note that by the extension of $A$ to $L_{2,\rho}({\mathbb{R}};H)$ is maximal monotone for each $\rho\in{\mathbb{R}}.$ We will denote this extension by $\mathcal{A}_{\rho}.$

First, we show that imply for the operators $\mathcal{M}_{\rho},\mathcal{N}_{\rho},\mathcal{M}_{\rho}'$ and the relation $\mathcal{A}_{\rho}$ for each $\rho\geq\rho_{0}.$

\[lem:evo\_implies\_causal\]We assume . Then for each $\rho\geq\rho_{0}$ the operators $\mathcal{M}_{\rho},\mathcal{N}_{\rho},\mathcal{M}_{\rho}'$ and the relation $\mathcal{A}_{\rho}$ satisfy . Moreover, the constant $c$ in (b) is independent of $\rho$.

It is clear that $\mathcal{M}_{\rho},\mathcal{N}_{\rho},\mathcal{M}_{\rho}'$ satisfy (b). Moreover, the constant $c$ in (b) is the same as in (b) and thus, independent of $\rho.$ To show (a), let $u\in{\operatorname{dom}}(\partial_{t,\rho}).$ We need to show that $\mathcal{M}_{\rho}u\in{\operatorname{dom}}(\partial_{t,\rho})$ and $$\partial_{t,\rho}\mathcal{M}_{\rho}u=\mathcal{M}_{\rho}\partial_{t,\rho}u+\mathcal{M}_{\rho}'u.$$ Choose a sequence $(\varphi_{n})_{n\in{\mathbb{N}}}$ in $C_{c}^{\infty}({\mathbb{R}};H)$ such that $\varphi_{n}\to u$ in $H_{\rho}^{1}({\mathbb{R}};H).$ Note that by (a) we have that $$\left(\mathcal{M}\varphi_{n}\right)'=\mathcal{M}\varphi'_{n}+\mathcal{M}'\varphi_{n}\in L_{2,\rho}({\mathbb{R}};H)\quad(n\in{\mathbb{N}})$$ and hence, $\mathcal{M}\varphi_{n}\in{\operatorname{dom}}(\partial_{t,\rho})$ (compare (a)). Moreover, $$\partial_{t,\rho}\mathcal{M}\varphi_{n}=\mathcal{M}\varphi'_{n}+\mathcal{M}'\varphi_{n}\to\mathcal{M}_{\rho}\partial_{t,\rho}u+\mathcal{M}'_{\rho}u\quad(n\to\infty)$$ in $L_{2,\rho}({\mathbb{R}};H)$ and since also $\mathcal{M}\varphi_{n}\to\mathcal{M}_{\rho}u$, the assertion follows by the closedness of $\partial_{t,\rho}.$ It 