Let $(A,\mathcal{D}(A))$ be an infinitesimal generator of a strongly continuous semigroup $(T(t))_{t\ge 0}$ on a Banach space $X$ and define on $\mathcal{X} := X \times X$ the operator matrix $$\mathcal{A}=\left( \begin{array}{cc} 0 & A \\ A & 0 \\ \end{array} \right)$$

with domain $\mathcal{D}(\mathcal{A}) := \mathcal{D}(A) \times D(A).$

I want to know if $\mathcal{A}$ generates a strongly continuous semigroup on the product space $\mathcal{X}$.