Criteria for eventual domination of operator semigroups and resolvents

TL;DR

This paper characterizes when one operator semigroup eventually dominates another on Banach lattices, explores related resolvent dominance near spectral bounds, and applies findings to differential operators and positivity principles.

Contribution

It provides new theoretical criteria for eventual domination of semigroups and resolvents, extending previous positivity-focused results to more general settings.

Findings

Characterization of eventual domination between semigroups

Analysis of resolvent dominance near spectral bounds

Applications to differential operators and positivity principles

Abstract

We consider two $C_0$-semigroups $(e^{tA})_{t \ge 0}$ and $(e^{tB})_{t \ge 0}$ on function spaces (or, more generally, on Banach lattices) and analyse eventual domination between them in the sense that $|e^{tA}f| \le e^{tB}|f|$ for all sufficiently large times $t$. We characterise this behaviour and prove a number of theoretical results which complement earlier results given by Mugnolo and the second author in the special case where both semigroups are positive for large times. Moreover, we study the analogous question of whether the resolvent of $B$ eventually dominates the resolvent of $A$ close to the spectral bound of $B$. This is closely related to the so-called maximum and anti-maximum principles. In order to demonstrate how our results can be used, we include several applications to concrete differential operators. At the end of the paper, we demonstrate that eventual positivity of the resolvent of a semigroup generator is closely related to eventual positivity of the Ces\`aro means of the associated semigroup.