Optimal decay of semi-uniformly stable operator semigroups with empty spectrum

TL;DR

This paper demonstrates that the decay rate of semi-uniformly stable operator semigroups cannot be determined solely from the spectrum of their generators, by constructing examples with empty spectrum but arbitrary decay behavior.

Contribution

It constructs specific semigroups with empty spectrum that defy decay rate quantification based on spectral information alone.

Findings

Decay rates cannot be inferred solely from the spectrum.

Existence of semigroups with empty spectrum and arbitrary decay behavior.

Spectral data is insufficient for decay rate estimation.

Abstract

We show that it is impossible to quantify the decay rate of a semi-uniformly stable operator semigroup based on sole knowledge of the spectrum of its infinitesimal generator. More precisely, given an arbitrary positive function $r$ vanishing at $\infty$, we construct a Banach space $X$ and a bounded semigroup $ (T(t))_{t \geq 0}$ of operators on it whose infinitesimal generator $A$ has empty spectrum $\sigma(A)=\varnothing$, but for which, for some $x \in X$, $$ \limsup_{t\to\infty} \frac{\|T(t)A^{-1}x\|_{X}}{r(t)}=\infty. $$