Analyticity of positive semigroups is inherited under domination

TL;DR

This paper proves that if a positive semigroup T is analytic and dominates another semigroup S on a Banach lattice, then S is also analytic, resolving an open problem from 2004.

Contribution

It establishes that analyticity is inherited under domination for positive semigroups, using spectral theory of positive operators.

Findings

Analyticity of T implies analyticity of S under domination.

Spectral theory of positive operators is key to the proof.

Answers an open problem posed by Arendt in 2004.

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.