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

TL;DR

The paper shows that decay rates for semi-uniformly stable C0-semigroups cannot be quantified from the spectrum of the generator alone. It constructs a bounded C0-semigroup with empty spectrum yet with divergent normalized orbit growth relative to any vanishing rate r, via a left translation semigroup on a carefully designed Banach space whose norm encodes Laplace-transform control. The arguments leverage Ingham-Karamata-type Tauberian ideas, connecting the spectral location to quantitative decay and highlighting the necessity of resolvent-type information for precise decay rates. This clarifies fundamental limitations in spectral-based decay estimates and informs strategies for obtaining quantitative stability results.

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 $σ(A)=\varnothing$, but for which, for some $x \in X$, $$ \limsup_{t\to\infty} \frac{\|T(t)A^{-1}x\|_{X}}{r(t)}=\infty. $$

Theorems & Definitions (6)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: C-N-V-NoteDensFuncEntLaplaceTrans
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['t:ExampleAbsenceOfRemainder']}