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If $E$ is a $\mathbb R$-Banach space, $(T(t))_{t\ge0}$ is a semigroup of bounded linear operators on $E$ and $(\mathcal D(A),A)$ denotes the generator of $(T(t))_{t\ge0}$, is $(\mathcal D(A),A)$ closed? I know that this is true and how we prove it, when $(T(t))_{t\ge0}$ is strongly continuous. Is there a counterexample if $(T(t))_{t\ge0}$ is not strongly continuous?

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