Let $\Omega$ be an open bounded set of $\mathbb{R}^N$ with smooth boundary. Consider a strongly continuous (or $C_0$-) semigroup of operators $(T(t))_{t\geq 0}$ defined on the Hilbert space $L^2(\Omega)$ and $(D(A),A)$ its infinitesimal generator. Assume that there exist a constant $\omega>0$ and $M$ such that $$\left|T(t)f\right|_{L^2(\Omega)}\leq Me^{-\omega t}\left|f\right|_{L^2(\Omega)} \ \ \ \ \ \ \ \ \ (1)$$ for all $f\in D(A)$, then by density of $D(A)$ in $L^2(\Omega)$ we have $$\left|T(t)\right|_{\text{operator norm}}\leq Me^{-\omega t}.$$ Now if we replace $(1)$ by the apparently weaker assumption:
$$\left|T(t)f\right|_{L^2(\Omega)}\leq Me^{-\omega t}\left|f\right|_{D(A)} \ \ \ \ \ \ \ \ \ (2)$$ for all $f\in D(A)$, where $\left|f\right|_{D(A)}:=\left|f\right|_{L^2(\Omega)}+\left|Af\right|_{L^2(\Omega)}$ is the graph norm. Do we get the same conclusion?