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Locally Eventually Positive Operator Semigroups

Sahiba Arora

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.

Locally Eventually Positive Operator Semigroups

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 -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 and the Dirichlet bi-Laplacian on -spaces. Besides, we establish various spectral and convergence properties of locally eventually positive semigroups.

Paper Structure

This paper contains 7 sections, 26 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Terminology and preliminaries
  3. Sufficient conditions for individual local eventual positivity
  4. Sufficient conditions for uniform local eventual positivity
  5. Applications
  6. Consequences of local eventual positivity I
  7. Consequences of local eventual positivity II

Key Result

Proposition 3.1

Let Assumptions ass:standing-assumptions be fulfilled. Suppose $\lambda_0 \in \mathbb{R}$ is a spectral value of $A$ and a simple pole of the resolvent $\mathcal{R}(\mathord{\,\cdot\,},A)$ such that the eigenspace ${\ker(\lambda_0-A)}$ is one-dimensional. Let $x$ be a non-zero vector in $\ker(\lambd

Theorems & Definitions (52)

  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof : Proof of Theorem \ref{['thm:sufficient-individual-resolvent']}
  • Theorem 3.3
  • proof : Proof of Theorem \ref{['thm:sufficient-individual-semigroup-resolvent']}
  • Corollary 3.5
  • ...and 42 more