# Stability of densly defined $C_{0}$-semigroup

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.

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 $$\mathbb{C}$$.