On the extension of battys theorem on the semigroup asymptotic stability

TL;DR

This paper extends Batty's theorem on the asymptotic behavior of $C_0$-semigroups, providing new conditions under which the semigroup norm times the inverse of the generator tends to zero, even for unbounded semigroups with spectra outside the left-half plane.

Contribution

It generalizes Batty's theorem by establishing broader conditions for the asymptotic decay of semigroup norms, including unbounded cases with less restrictive spectral assumptions.

Findings

The property holds for semigroups with sufficiently regular norms.

Examples of unbounded semigroups with spectra outside the left-half plane exhibit the property.

A new sufficient condition for the property to hold is provided.

Abstract

The well-known Batty's theorem states that if a $C_0$-semigroup $T(t)$ is bounded and the spectrum of the generator $A$ is contained in the open left-half plane of $\mathbb{C}$, then $\|T(t)A^{-1}\|$ tends to $0$. This can be thought of as a particular case of a more general property that, for $\omega_0>-\infty$ and $(\omega_0+i\mathbb{R})\cap \sigma(A)=\emptyset$ it holds $\|T(t)(A-\omega_0 I)^{-1}\|/\|T(t)\|$ tends to 0. We show that it is true for $\|T(t)\|$ regular enough, however we give examples of unbounded semigroups, with the spectrum of the generator not contained in the open left-half plane of $\mathbb{C}$, with the above property. Moreover we give a more general sufficient condition for this property to hold, thus extending Batty's theorem.