A Note on the Krein-Rutman Theorem for Sectorial Operators

TL;DR

This paper generalizes the Krein-Rutman theorem for sectorial operators, providing tools applicable to elliptic operators and insights into the eigenspaces related to non-principal eigenvalues, aiding the study of evolution equations.

Contribution

It introduces generalized versions of the Krein-Rutman theorem tailored for sectorial operators, including information on eigenspaces for non-principal eigenvalues.

Findings

Generalized Krein-Rutman theorems for sectorial operators

Application-ready formulations for elliptic operators

Enhanced understanding of eigenspaces for non-principal eigenvalues

Abstract

In this note we present some generalized versions of the Krein-Rutman theorem for sectorial operators. They are formulated in a fashion that can be easily applied to elliptic operators. Another feature of these generalized versions is that they contain some information on the generalized eigenspaces associated with non-principal eigenvalues, which are helpful in the study of the dynamics of evolution equations in ordered Banach spaces.