# The role of domination and smoothing conditions in the theory of eventually positive semigroups

## TL;DR

This paper investigates how domination and smoothing conditions influence the spectral theory of eventually positive semigroups, revealing their implications for compactness and eigenvector existence.

## Contribution

It demonstrates that domination and smoothing conditions imply compactness on many function spaces and can be omitted in key spectral theorems, advancing the understanding of positive operator semigroups.

## Key findings

- Domination and smoothing imply compactness properties.
- Conditions can be omitted in Perron--Frobenius type theorems.
- Existence of positive eigenvectors under eventual positivity.

## Abstract

We perform an in-depth study of some domination and smoothing properties of linear operators and of their role within the theory of eventually positive operator semigroups. On the one hand we prove that, on many important function spaces, they imply compactness properties. On the other hand, we show that these conditions can be omitted in a number of Perron--Frobenius type spectral theorems. We furthermore prove a Kre\u{\i}n--Rutman type theorem on the existence of positive eigenvectors and eigenfunctionals under certain eventual positivity conditions.