that ${\operatorname{dom}}\left((\partial_{t,\rho}\mathcal{M}+\mathcal{N}+\mathcal{A})^{-1}\right)$ is dense in $L_{2,\rho}({\mathbb{R}};H).$ Hence, there exist sequences $(f_{n})_{n\in{\mathbb{N}}}$ and $(g_{n})_{n\in{\mathbb{N}}}$ in ${\operatorname{dom}}\left((\partial_{t,\rho}\mathcal{M}+\mathcal{N}+\mathcal{A})^{-1}\right)$ such that $f_{n}\to f$ and $g_{n}\to g$ in $L_{2,\rho}({\mathbb{R}};H).$ Setting $$u_{n}\coloneqq\mathcal{S}_{\rho}(f_{n})\text{ and }v_{n}\coloneqq\mathcal{S}_{\rho}(g_{n})$$ for $n\in{\mathbb{N}}$, it follows that $u_{n}\to u$ and $v_{n}\to v$ in $L_{2,\rho}({\mathbb{R}};H)$ by continuity of $\mathcal{S}_{\rho}.$ By definition, for $n\in{\mathbb{N}}$ there exist $x_{n},y_{n}\in L_{2,\rho}({\mathbb{R}};H)$ such that $(u_{n},x_{n}),(v_{n},y_{n})\in\mathcal{A}$ and $$\begin{aligned}
\partial_{t,\rho}\mathcal{M}u_{n}+\mathcal{N}u_{n}+x_{n} & =f_{n}\\
\partial_{t,\rho}\mathcal{M}v_{n}+\mathcal{N}v_{n}+y_{n} & =g_{n}.\end{aligned}$$ Moreover, by and the monotonicity of $\chi_{{\mathbb{R}}_{\leq a}}\mathcal{A}$ we estimate $$\begin{aligned}
 & {\operatorname{Re}}\langle f_{n}-g_{n},\chi_{{\mathbb{R}}_{\leq a}}(u_{n}-v_{n})\rangle_{\rho}\\
 & ={\operatorname{Re}}\langle\left(\partial_{t,\rho}\mathcal{M}+\mathcal{N}\right)(u_{n}-v_{n}),\chi_{\mathbb{R}_{\leq a}}(u_{n}-v_{n})\rangle_{\rho}+{\operatorname{Re}}\langle x_{n}-y_{n},\chi_{\mathbb{R}_{\leq a}}(u_{n}-v_{n})\rangle_{\rho}\\
 & \geq c|\chi_{\mathbb{R}_{\leq a}}(u_{n}-v_{n})|_{\rho}^{2}.\end{aligned}$$ Letting $n$ tend to infinity, it follows that $$0={\operatorname{Re}}\langle f-g,\chi_{{\mathbb{R}}_{\leq a}}(u-v)\rangle_{\rho}\geq c|\chi_{\mathbb{R}_{\leq a}}(u-v)|_{\rho}^{2},$$ which proves $u=v$ on $]-\infty,a]$ and hence, the causality of $\mathcal{S}_{\rho}$ follows.

In order to formulate the independence result we need the concept of evolutionary mappings. These mappings were introduced in [@Wauricl2014; @Waurick_habil], however in a more general way than needed here.

Let $\rho_{0}>0$. A linear mapping $T:C_{c}^{\infty}({\mathbb{R}};H)\to\bigcap_{\mu\geq\rho_{0}}L_{2,\mu}({\mathbb{R}};H)$ is called *evolutionary at $\rho_{0}$,* if for each $\rho\geq\rho_{0}$ the mapping $$T:C_{c}^{\infty}({\mathbb{R}};H)\subseteq L_{2,\rho}({\mathbb{R}};H)\to L_{2,\rho}({\mathbb{R}};H)$$ is bounded and $\|T\|_{\mathrm{ev},\rho_{0}}\coloneqq\sup_{\rho\geq\rho_{0}}\|T\|_{L(L_{2,\rho}({\mathbb{R}};H))}<\infty$. If $T$ is evolutionary at $\rho_{0}$, we denote the (unique) extension of $T$ to $L_{2,\rho}({\mathbb{R}};H)$ by $T_{\rho}$ for each $\rho\geq\rho_{0}.$

We are now ready to formulate our hypotheses, which will allow us to prove 