Growth rate of eventually positive Kreiss bounded $C_0$-semigroups on $L^p$ and $\mathcal{C}(K)$

Abstract

In this paper, we compare several Cesàro and Kreiss type boundedness conditions for a $C_0$-semigroup on a Banach space and we show that those conditions are all equivalent for a positive semigroup on a Banach lattice. Furthermore, we give an estimate of the growth rate of a Kreiss bounded and eventually positive $C_0$-semigroup $(T_t{)}_{t\ge 0}$ on certain Banach lattices $X$. We prove that if $X$ is an $L^p$-space, $1<p<+\infty$, then $\Vert T_t\Vert =\mathcal{O}\left(t/\log (t{)}^{\max (1/p,1/p^{\prime })}\right)$ and if $X$ is an $(\text{AL})$ or $(\text{AM})$-space, then $\Vert T_t\Vert =\mathcal{O}(t^{1-\epsilon })$ for some $\epsilon \in (0,1)$, improving previous estimates.

v2

References in corpus (1)

• Stability of (eventually) positive semigroups on spaces of continuous functions