For an operator $(A,\mathcal{D}(A))$ on a Banach space $X$ define on $\mathcal{X} := X \times X$ the operator matrix $$\mathcal{A}=\left( \begin{array}{cc} A & 0 \\ 0 & A \\ \end{array} \right)$$
with domain $\mathcal{D}(\mathcal{A}) := \mathcal{D}(A) \times D(A).$
I want to show that the following assertions are equivalent.
(i) $A$ generates a strongly continuous semigroup on $X$.
(ii) $\mathcal{A}$ generates a strongly continuous semigroup on $\mathcal{X}$.