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

Abstract

We prove that a Kreiss bounded $$C_0$$ C 0 semigroup $$(T_t)_{t \ge 0}$$ ( T t ) t ≥ 0 on a Hilbert space has asymptotics $$\left\| T_t\right\| = {\mathcal{O}}\big (t/\sqrt{\log (t)}\big ).$$ T t = O ( t / log ( t ) ) . Then, we give an application to perturbed wave equation.