Work

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

Year: 2025

Type: article

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 r vanishing at ∞ \infty , we construct a Banach space X X and a bounded semigroup ( T ( t ) ) t ≥ 0 (T(t))_{t \geq 0} of operators on it whose infinitesimal generator A A has empty spectrum σ ( A ) = ∅ \sigma (A)=\varnothing , but for which, for some x ∈ X x \in X , lim sup t → ∞ ‖ T ( t ) A − 1 x ‖ X r ( t ) = ∞ . \begin{equation*} \limsup _{t\to \infty } \frac {\|T(t)A^{-1}x\|_{X}}{r(t)}=\infty . \end{equation*}

Source: Proceedings of the American Mathematical Society

DOI: 10.1090/proc/17466

Authors: Morgan Callewaert, Lenny Neyt, Jasson Vindas

Institutions: Ghent University, University of Vienna

Language: English

Topic: Stability and Controllability of Differential Equations

Subfield: Control and Systems Engineering

Field: Engineering

Domain: Physical Sciences

Open Access status: hybrid

Funders: Austrian Science Fund, Universiteit Gent

Awards: G3035029418, G705532438, G7218683284, G9000263684