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.
1 Answer
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 $\mathbb{C}$.


$\begingroup$ @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" $\endgroup$ Feb 9, 2021 at 3:11