is left to prove that $\mathcal{A}_{\rho}$ satisfies . We had already remarked that $\mathcal{A}_{\rho}$ is maximal monotone by . Moreover, for $(u,v)\in\mathcal{A}_{\rho}$ and $h\geq0$ we have that $$\begin{aligned}
(u,v)\in\mathcal{A}_{\rho} & \Rightarrow(u(t),v(t))\in A\quad(t\in{\mathbb{R}}\text{ a.e.})\\
 & \Rightarrow(u(t+h),v(t+h))\in A\quad(t\in{\mathbb{R}}\text{ a.e.})\\
 & \Rightarrow(\tau_{h}u,\tau_{h}v)\in\mathcal{A}_{\rho}.\end{aligned}$$ Moreover, for $a\in{\mathbb{R}}$ and $(u,v),(x,y)\in\mathcal{A}_{\rho}$ we have that $${\operatorname{Re}}\langle u-x,\chi_{{\mathbb{R}}_{\leq a}}(v-y)\rangle_{\rho}=\intop_{-\infty}^{a}{\operatorname{Re}}\langle u(t)-x(t),v(t)-y(t)\rangle{\mathrm{e}}^{-2\rho t}\text{ d}t\geq0,$$ since $(u(t),v(t)),(x(t),y(t))\in A$ for almost every $t\in{\mathbb{R}}$ and $A$ is monotone.

In order to prove that the solution operator $\mathcal{S}_{\rho}=\left(\overline{\partial_{t,\rho}\mathcal{M}_{\rho}+\mathcal{N}_{\rho}+\mathcal{A}_{\rho}}\right)^{-1}$ is independent of $\rho$ in the sense that for each $\mu,\rho\geq\rho_{0}$ $$\mathcal{S}_{\rho}(f)=\mathcal{S}_{\mu}(f)\quad(f\in L_{2,\rho}({\mathbb{R}};H)\cap L_{2,\mu}({\mathbb{R}};H))$$ we prove this property for the solution operator $\mathcal{S}_{\rho,\mathrm{aux}}=\left(\overline{\partial_{t,\rho}\mathcal{M}_{\rho}-\mathcal{M}_{\rho}'+\delta+\mathcal{A}_{\rho}}\right)^{-1}$ with $\delta>\|\mathcal{M}'\|_{\mathrm{ev},\rho_{0}}+\|\mathcal{N}\|_{\mathrm{ev},\rho_{0}}$ first. For doing so, we need to show that the solution operator $\mathcal{S}_{\rho,\mathrm{aux}}$ is causal. This will be shown in the next proposition. However, we first need the following lemma, showing an important property of evolutionary mappings.

\[lem:causality\_evo\]Let $\rho_{0}>0$ and $T:C_{c}^{\infty}({\mathbb{R}};H)\to\bigcap_{\mu\geq\rho_{0}}L_{2,\mu}({\mathbb{R}};H)$ evolutionary at $\rho_{0}.$ Then, for each $\mu,\rho\geq\rho_{0}$ and $f\in L_{2,\rho}({\mathbb{R}};H)\cap L_{2,\mu}({\mathbb{R}};H)$ we have that $$T_{\mu}f=T_{\rho}f.$$ Moreover, for each $\rho\geq\rho_{0}$ the mapping $T_{\rho}$ is causal.

The first assertion is clear, since a function $f\in L_{2,\rho}({\mathbb{R}};H)\cap L_{2,\mu}({\mathbb{R}};H)$ can be approximated by one sequence $(\varphi_{n})_{n\in{\mathbb{N}}}$ in $C_{c}^{\infty}({\mathbb{R}};H)$ such that $\varphi_{n}\to f$ in $L_{2,\mu}({\mathbb{R}};H)$ and $L_{2,\rho}({\mathbb{R}};H)$, see e.g. [@Trostorff2013_stability Lemma 3.5]. For proving the causality 