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arXiv · 2207.08443

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

L. Arnold·C. Coine
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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 $\|T_t\| = \mathcal{O}\left(t/\log(t)^{\max(1/p,1/p')}\right)$ and if $X$ is an $(\text{AL})$ or $(\text{AM})$-space, then $\|T_t\|=\mathcal{O}(t^{1-ε})$ for some $ε\in (0,1)$, improving previous estimates.

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L. Arnold, C. Coine. 2022-11-22. Growth rate of eventually positive Kreiss bounded $C_0$-semigroups on $L^p$ and $\mathcal{C}(K)$. https://arxiv.org/abs/2207.08443

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