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

In this paper, we compare several Cesàro- and Kreiss-type boundedness conditions for a \(C_0\)-semigroup on Banach space show that those are all equivalent positive semigroup lattice. Furthermore, give an estimate of the growth rate Kreiss bounded eventually \((T_t)_{t\ge 0}\) certain lattices X. We prove if X is \(L^p\)-space, \(1<p<+\infty \), then \(\Vert T_t\Vert = \mathcal {O}\left( t/\log (t)^{\max (1/p,1/p')}\right) \((\text {AL})\) or {AM})\)-space, =\mathcal {O}(t^{1-\epsilon })\) some \(\epsilon \in (0,1)\), improving previous estimates.