Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Analyticity of positive semigroups is inherited under domination

Jochen Glück

Abstract

For positive $C_0$-semigroups $S$ and $T$ on a Banach lattice such that $S(t) \le T(t)$ for all times $t$, we prove that analyticity of $T$ implies analyticity of $S$. This answers an open problem posed by Arendt in 2004. Our proof is based on a spectral theoretic argument: we apply spectral theory of positive operators to multiplication operators that are induced by $S$ and $T$ on a vector-valued function space.

Analyticity of positive semigroups is inherited under domination

Abstract

For positive -semigroups and on a Banach lattice such that for all times , we prove that analyticity of implies analyticity of . This answers an open problem posed by Arendt in 2004. Our proof is based on a spectral theoretic argument: we apply spectral theory of positive operators to multiplication operators that are induced by and on a vector-valued function space.
Paper Structure (1 section, 5 theorems, 9 equations)

This paper contains 1 section, 5 theorems, 9 equations.

Table of Contents

  1. Acknowledgement

Key Result

Theorem 1

Let $S,T$ be $C_0$-semigroups on a complex Banach lattice $E$ such that $0 \le S(t) \le T(t)$ for all $t \in [0,\infty)$. If $T$ is analytic, then so is $S$.

Theorems & Definitions (8)

  • Theorem 1
  • Proposition 2: Kato
  • Proposition 3: Räbiger--Wolff
  • Lemma 4
  • proof : Proof of Lemma \ref{['lem:invertible-multiplication-op']}
  • Corollary 5
  • proof : Proof of Theorem \ref{['thm:intro-dom']}
  • Remark 6