Let $(S(t))_{t \geq 0}$ be a $C_{0}$-semigroup on $H$ where $H$ is a Hilbert space. Suppose that $(S(t))_{t \geq 0}$ satisfies the following estimate on a dense subspace on $H$ $$||S(t)x||_H \leq e^{-t}||x||_H.$$ Can this estimate be extended for any $x \in H$?. Thank you.
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Yes, let $x \in H$ and $D$ be the dense subspace and $(x_n) \subset D$ such that $x_n \to x$ in $H$, since $S(t) \in L(H,H)$ then $S(t)x_n \to S(t)x$ in $H$. Therefore, $\|S(t)x_n\| \to \|S(t)x\|$. Passing to the limit the inequality $\|S(t)x_n\|_H \leq e^{-t}\|x_n\|_{H}$ we obtain the desired since $\|x_n\| \rightarrow \|x\|$ in $\mathbb{C}$.
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$\begingroup$ @Gustave If you are satisfied with the answer, then consider clicking the option to "accept" it, so that the question does not show up as "unanswered" $\endgroup$ Commented Feb 9, 2021 at 3:11