A polynomial gap below linear growth of Kreiss bounded $C_0$-semigroups on Hilbert spaces

We prove that every Kreiss bounded $C_0$-semigroup $(T_t)_{t\geq0}$ on a Hilbert space satisfies $$\|T_t\|\leq C(1+t)^{1-\varepsilon_K}, \qquad t\geq0,$$ where $\varepsilon_K>0$ depends explicitly only on the Kreiss constant. This improves the previously known estimate $O(t/\sqrt{\log(t+1)})$ and shows that every Kreiss bounded $C_0$-semigroup has a genuine polynomial gap below linear growth. In view of the examples of Eisner and Zwart with growth arbitrarily close to linear, no universal positive exponent can hold for the whole class of Kreiss bounded $C_0$ semigroups on Hilbert spaces.