 Contraction semigroup

A contraction semigroup is a strongly continuous semigroup of bounded linear operators satisfying $\Vert S(t)\Vert \le 1$ for every $t\ge 0$. The Hille-Yosida theorem characterizes its infinitesimal generator of a semigroup by dense domain, closedness, and the resolvent estimates $\Vert (\lambda -A{)}^{-n}\Vert \le {\lambda }^{-n}$ for real $\lambda >0$ and integers $n\ge 1$.