Polynomial gaps below linear growth for Kreiss bounded semigroups and operators

Abstract

We prove that every Kreiss bounded $C_0$-semigroup $(T_t{)}_{t\ge 0}$ on a Hilbert space satisfies \[ \|T_t\|\le C(1+t)^{1-\varepsilon_K}, \qquad t\ge0, \] where ${\epsilon }_K>0$ 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. We further obtain a non-quantitative polynomial gap for individually eventually positive Kreiss bounded $C_0$-semigroups on $L^p$-spaces. Finally we prove that every Kreiss bounded operator on a UMD Banach space has a polynomial gap below linear growth

Expanded version: Added a new section on individually eventually positive Kreiss bounded C0-semigroups on Lp-spaces, establishing a non-quantitative polynomial gap below linear growth. Added a new section proving a polynomial growth gap for Kreiss bounded operators on UMD spaces. 21 pages