Locally Eventually Positive Operator Semigroups

TL;DR

This paper develops a new theory of locally eventually positive operator semigroups on Banach lattices, relaxing previous eigenvector positivity requirements to include a broader class of examples and analyzing their spectral and convergence properties.

Contribution

It introduces sufficient criteria for local eventual positivity without the need for strongly positive eigenvectors, expanding applicability to operators like the Laplacian and bi-Laplacian.

Findings

Broader class of locally eventually positive semigroups identified.

Criteria established for individual and uniform local eventual positivity.

Spectral and convergence properties of these semigroups analyzed.

Abstract

We initiate a theory of locally eventually positive operator semigroups on Banach lattices. Intuitively this means: given a positive initial datum, the solution of the corresponding Cauchy problem becomes (and stays) positive in a part of the domain, after a sufficiently large time. A drawback of the present theory of eventually positive $C_0$-semigroups is that it is applicable only when the leading eigenvalue of the semigroup generator has a strongly positive eigenvector. We weaken this requirement and give sufficient criteria for individual and uniform local eventual positivity of the semigroup. This allows us to treat a larger class of examples by giving us more freedom on the domain when dealing with function spaces -- for instance, the square of the Laplace operator with Dirichlet boundary conditions on $L^2$ and the Dirichlet bi-Laplacian on $L^p$-spaces. Besides, we establish various spectral and convergence properties of locally eventually positive semigroups.