Let $T_t:L\to L$ be a semigroup of linear operators $T_t$ acting on a Banach space $L$. Assume that $ \|T_t\| := \sup\limits_{f\in L}\frac{\|T_tf\|_L}{\|f\|_L} \leq 1 $ for all $t\geq 0$. The infinitesimal operator is given by $ \mathcal Af = \lim\limits_{h\to 0}\frac{T_hf-f}{h} $ for any $f\in\mathcal D_{\mathcal A}$, i.e. for each $f$ such that this limit exists. $\mathcal A$ is clearly linear, and my questions are
if it is necessary bounded on $\mathcal D_{\mathcal A}$?
If it's not, could you please give an example when $\mathcal A$ is unbounded?
If there is an example when \|T_t\|'\leq 1 and \|T_t\|''\leq 1 for two different norms \|\cdot\|'_L and \|\cdot\|''_L but \|\mathcal A\|'<\infty while \|\mathcal A\|'' =\infty?
Edited: in the view of the answers, it's still unclear with a part 3. If given one norm on $L$ the linear operator $\mathcal A$ is bounded is it necessary that it will be bounded in another norm on $L$? E.g. in which norm Laplacian $\Delta$ is unbounded and why it is impossbile to give a norm on $C^2$ which will make it bounded?