Let $\mathcal{H}$ be Hilbert space of the vectors $f = \left(% \begin{array}{c} f^{+}(x) \\ f^{-}(x) \\ \end{array}% \right)$ where $f^{+}(x),$ $f^{-}(x)$ belong to the class $L^{2}(-1,1)$ of the functions which are square integrable on $(-1,1)$. The scalar product of two elements $f$, $g \in \mathcal{H}$ is defined by$$\displaystyle{<f,g> = \int_{-1}^{1}\left[f^{+}(x)g^{+}(x)+f^{-}(x)g^{-}(x)\right]dx}$$
while the norm is $\|f\| = <f,f>^{\frac{1}{2}}$. Let $s$ be a positive constant and $A$ the following operator $$A:=\left(% \begin{array}{cc} -\displaystyle{\frac{\partial}{\partial x}} & \displaystyle{\frac{s}{2}} \\ \\ \displaystyle{\frac{s}{2}} & \displaystyle{\frac{\partial}{\partial x}} \\ \end{array}% \right)$$
with domain $$\mathcal{D}(A) :=\left\{f = \left(% \begin{array}{c} f^{+}(x) \\ f^{-}(x) \\ \end{array}% \right)\in\mathcal{H}\,\mbox{such that},\begin{aligned}\,1)&\,f^{+}\,\mbox{and}\,f^{-}\,\mbox{are absolutely continuous},\\2)&\,f^{+}(-1) = f^{-}(1) = 0\,\,\mbox{and}\\3)&\,Af \in\mathcal{H}\end{aligned}\right\}.$$
My question is: How we can prove that the operator $A$ generates $C_{0}$ semigroup on $\mathcal{H}$?