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

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 $(T(t){)}_{t\ge 0}$ of operators on it whose infinitesimal generator $A$ has empty spectrum $\ddot{I}ƒ(A)=\varnothing$, but for which, for some $x\in X$, $$\underset{t\to \infty }{\mathrm{limsup}}\frac{\Vert T(t)A^{-1}x{\Vert }_X}{r(t)}=\infty .$$

8 pages