Refined scales of decaying rates of operator semigroups on Hilbert spaces: typical behaviour

TL;DR

This paper investigates how the decay rates of operator semigroups on Hilbert spaces relate to spectral properties of their generators, revealing that for stable but not exponentially stable semigroups, decay behavior varies with sequences of time.

Contribution

It establishes typical decay rate behaviors of semigroups based on spectral properties, especially for those that are stable but not exponentially stable.

Findings

Decay rates depend on sequences of time tending to infinity.

Stable but not exponentially stable semigroups exhibit variable decay behaviors.

Spectral properties influence the typical decay rates of semigroup orbits.

Abstract

We study relations between the decaying rates of operator semigroups on Hilbert spaces and some spectral properties of their respective generators; in particular, we show that the decaying rates of orbits of semigroups which are stable but not exponentially stable, typically in Baire's sense, depend on sequences of time going to infinity.