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

TL;DR

This paper investigates the growth rates of eventually positive Kreiss bounded $C_0$-semigroups on Banach lattices, providing new bounds for $L^p$-spaces and AL/AM-spaces, and establishing equivalences of boundedness conditions.

Contribution

It establishes the equivalence of Cesàro and Kreiss boundedness conditions for positive semigroups and derives improved growth rate estimates for such semigroups on specific Banach lattices.

Findings

For $L^p$-spaces, $ orm{T_t} = igo(t/ ext{log}(t)^{ ext{max}(1/p,1/p')})$.

For AL or AM spaces, $ orm{T_t} = igo(t^{1- ext{epsilon}})$ for some $ ext{epsilon} ext{ in } (0,1)$.

Boundedness conditions are equivalent for positive semigroups on Banach lattices.

Abstract

In this paper, we compare several Ces\`aro 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 $\|T_t\| = \mathcal{O}\left(t/\log(t)^{\max(1/p,1/p')}\right)$ and if $X$ is an $(\text{AL})$ or $(\text{AM})$-space, then $\|T_t\|=\mathcal{O}(t^{1-\epsilon})$ for some $\epsilon \in (0,1)$, improving previous estimates.