Stability of Individually Eventually Positive Semigroups on $L^p$-Spaces

We answer a question raised by Vogt by proving that the growth bound of an individually eventually positive $C_0$-semigroup on an $L^p$-space ($1<p<\mathrm{\infty }$) coincides with the spectral bound of its generator. Our proof follows the general strategy of Vogt’s argument for the uniformly eventually positive case. The key new ingredient is a variant of an operator-range uniformisation principle of Arora and Glück, adapted here to time-dependent operator ranges. When applied to principal ideals, this principle turns individual eventual positivity into the uniform domination estimate required for Vogt’s stability argument.