Behavior of Kreiss bounded $C_0$-semigroups on a Hilbert space

Let $0<\alpha \le 1$. We prove that a $\alpha$-Kreiss bounded $C_0$ semigroup $(T_t{)}_{t\ge 0}$ on a Hilbert space has asymptotics $\mathrm{\mid }\mathrm{\mid }{T}_t\mathrm{\mid }\mathrm{\mid }=O\left(t^{\alpha }\mathrm{/}\sqrt{log(t)}\right)$. Then, we give an application to perturbed wave equation.