that it coincides with the causality defined above if $F$ is defined on the whole $L_{2,\rho}({\mathbb{R}};H)$. This was done in [@Waurick2013_causality].

We now state the hypotheses which allow us to deduce the causality of the solution operator.

\[hyp:causality\]Let $\mathcal{M},\mathcal{N},\mathcal{M}'\in L(L_{2,\rho}({\mathbb{R}};H))$ be such that:

(a) $\mathcal{M}\partial_{t,\rho}\subseteq\partial_{t,\rho}\mathcal{M}-\mathcal{M}',$

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

Moreover, let $\mathcal{A}\subseteq L_{2,\rho}({\mathbb{R}};H)\times L_{2,\rho}({\mathbb{R}};H)$ be maximal monotone such that $$\forall u,v\in L_{2,\rho}({\mathbb{R}};H):\,\left((u,v)\in\mathcal{A}\Rightarrow\forall h\geq0:(\tau_{h}u,\tau_{h}v)\in\mathcal{A}\right)$$ and assume that $\chi_{\mathbb{R}_{\leq a}}\mathcal{A}$ is monotone for each $a\in{\mathbb{R}}.$

We note that imply since (b) of follows from (b) by letting $a$ tend to infinity. Moreover, the monotonicity of $\chi_{\mathbb{R}_{\leq a}}\mathcal{A}$ states that for each $(u,v),(x,y)\in\mathcal{A}$ we have that $${\operatorname{Re}}\langle u-x,\chi_{\mathbb{R}_{\leq a}}(v-y)\rangle_{\rho}={\operatorname{Re}}\langle\chi_{\mathbb{R}_{\leq a}}(u-x),v-y\rangle_{\rho}\geq0.$$

\[lem:causality\_better\]Assume . Then $${\operatorname{Re}}\langle(\partial_{t,\rho}\mathcal{M}+\mathcal{N})u,\chi_{\mathbb{R}_{\leq a}}u\rangle_{\rho}\geq c|\chi_{\mathbb{R}_{\leq a}}u|_{\rho}^{2}$$ for each $u\in{\operatorname{dom}}(\partial_{t,\rho}\mathcal{M})$ and $a\in{\mathbb{R}}.$

By , $C_{c}^{\infty}({\mathbb{R}};H)$ is a core for $\partial_{t,\rho}\mathcal{M}.$ Hence, the assertion follows by approximating $u$ with elements in $C_{c}^{\infty}({\mathbb{R}};H)$ and noting that multiplication with $\chi_{{\mathbb{R}}_{\leq a}}$ is continuous on $L_{2,\rho}({\mathbb{R}};H).$

\[thm:causaliyt\] Assume . Then the solution operator $\mathcal{S}_{\rho}\coloneqq\left(\overline{\partial_{t,\rho}\mathcal{M}+\mathcal{N}+\mathcal{A}}\right)^{-1}$ from is causal.

Let $f,g\in L_{2,\rho}({\mathbb{R}};H)$ and assume that $f=g$ on $]-\infty,a]$ for some $a\in{\mathbb{R}}.$ We set $$u\coloneqq\mathcal{S}_{\rho}(f)\text{ and }v\coloneqq\mathcal{S}_{\rho}(g).$$ Since $\mathcal{S}_{\rho}=\left(\overline{\partial_{t,\rho}\mathcal{M}+\mathcal{N}+\mathcal{A}}\right)^{-1}=\overline{\left(\partial_{t,\rho}\mathcal{M}+\mathcal{N}+\mathcal{A}\right)^{-1}}$ is defined on the whole $L_{2,\rho}({\mathbb{R}};H)$ it follows 