1
$\begingroup$

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.

$\endgroup$

1 Answer 1

2
$\begingroup$

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 $\mathbb{C}$.

$\endgroup$
2
  • $\begingroup$ Thank you sir for your answer. $\endgroup$
    – Gustave
    Jan 25, 2021 at 20:52
  • $\begingroup$ @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" $\endgroup$
    – Yemon Choi
    Feb 9, 2021 at 3:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.