operator $\partial_{t,\rho}$ as the closure of the operator $$C_{c}^{\infty}({\mathbb{R}};H)\subseteq L_{2,\rho}({\mathbb{R}};H)\to L_{2,\rho}({\mathbb{R}};H):\,\phi\mapsto\phi',$$ where $C_{c}^{\infty}({\mathbb{R}};H)$ denotes the space of $H$-valued arbitrarily differentiable functions on ${\mathbb{R}}$ with compact support.

\[rem:derivative\]

(a) The domain of $\partial_{t,\rho}$ consists of those elements in $L_{2,\rho}({\mathbb{R}};H)$ whose distributional derivative lies again in $L_{2,\rho}({\mathbb{R}};H).$

(b) For $\rho=0$ the space $L_{2,0}({\mathbb{R}};H)$ is the usual $L_{2}$-space and the derivative $\partial_{t,0}$ coincides with the classical weak derivative with domain $H^{1}({\mathbb{R}};H).$

We recall some basic properties of $\partial_{t,\rho}$ and refer to [@Kalauch2011; @Picard2014_survey] for the respective proofs.

Let $\rho\geq0.$ Then the operator $\partial_{t,\rho}$ is normal with $\partial_{t,\rho}^{\ast}=-\partial_{t,\rho}+2\rho.$ Moreover, $\sigma(\partial_{t,\rho})=\left\{ {\mathrm{i}}t+\rho\,;\,t\in{\mathbb{R}}\right\} $. In particular, if $\rho>0,$ then $\partial_{t,\rho}$ is boundedly invertible with $\|\partial_{t,\rho}^{-1}\|=\frac{1}{\rho}$ and $$\left(\partial_{t,\rho}^{-1}f\right)(t)=\intop_{-\infty}^{t}f(s){\,\mathrm{d}}s\quad(f\in L_{2,\rho}({\mathbb{R}};H),t\in{\mathbb{R}}).$$

It should be noted that the bounded invertibility of $\partial_{t,\rho}$ just holds, since we deal with the whole real line and not just with the positive real line or an interval. The main reason is that on the positive real line or an interval, we need to impose initial conditions to obtain invertibility of $\partial_{t,\rho}$. By dealing with the whole real line, we implicitly impose a vanishing initial condition at $-\infty$. For the treatment of differential inclusions on $\mathbb{R}_{\geq0}$ within the framework introduced here, we refer to [@Trostorff2012_NA].

