Polynomial gaps below linear growth for Kreiss bounded semigroups and operators

Abstract

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. The same conclusion is obtained for positive Kreiss bounded $C_0$-semigroups on $L^p$-spaces, $1<p<\infty$, and in discrete time for Kreiss bounded operators on Hilbert spaces and positive Kreiss bounded operators on $L^p$-spaces. Finally, we obtain a non-quantitative polynomial gap for individually eventually positive Kreiss bounded $C_0$-semigroups on $L^p$-spaces.