Let $H$ be a separable complex Hilbert space, let $(e_i)_{i\in\mathbb{N}}$ be a complex othonormal vasis, and let $(\lambda_i)_{i\in\mathbb{N}}$ be a sequence of complex numbers s.t. $\sup_{i\in \mathbb{N}}Re(\lambda_i)<\infty$. Define the map $S:[0,\infty)\rightarrow \mathcal{L}^c(H)$ by $S(t)x=\sum_{i=1}^{\infty}e^{\lambda_i t}\langle e_i,x\rangle e_i$
for $x\in H$ and $t\geq 0$.
I was unsuccessfully trying to show that this is a strongly continuous semigroup of operators on $H$. Thank you for your help
Assuming you see how to show it's a semigroup of bounded operators and you're just stuck on the "strongly continuous" part:
We need to show that $||x-S(t)x||\to0$ as $t\to0$. Say $x=\sum a_ne_n$, where $\sum|a_n|^2<\infty$. Then $$||x-S(t)x||^2=\sum|1-e^{\lambda_nt}|^2|a_n|^2.$$The hypothesis on $\lambda_n$ shows that there exists $c$ with $$|e^{\lambda_nt}|\le c\quad(0\le t\le 1);$$hence $||x-S(t)x||^2\to0$ by dominated convergence (in $\ell^1$).