arXiv subjects

Morgan Callewaert

Publications and source records attributed to Morgan Callewaert.

3 recordsLinked to original sources

arXiv2026

Spectral characterizations of stable operator semigroups

We introduce the notion of local pseudofunction spectrum $σ_{PF}(A)$ for the infinitesimal generator $A$ of a bounded $C_0$-semigroup $\mathcal{T} = (T(t))_{t \geq 0}$ on a Banach space $X$ and show it is the right spectral concept to deliver a full characterization of the strong stability of $\mathcal{T}$: \[ \forall x \in X : ~ \lim_{t \to \infty} \| T(t) x \|_X = 0 \quad \Longleftrightarrow \quad σ_{PF}(A) = \varnothing. \] We demonstrate how this yields a quick proof of the well-known Arendt-Batty-Lyubich-Vũ theorem and establish novel stability results for semigroups whose local pseudofunction spectra are contained in a null subset of the imaginary axis through localization of range density conditions to the potential null singular set. We also obtain similar stability characterization theorems for individual orbits and for semi-uniform stability. As an application of our results, we provide spectral characterizations of asymptotically almost periodic $C_0$-semigroups with countable local pseudofunction spectrum. In addition, we prove optimal Tauberian theorems of Katznelson-Tzafriri type and discuss connections with Wiener kernels.

math.FA↗

arXiv2025

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

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

math.FA↗

arXiv2025

On a space of functions with entire Laplace transforms and its connection with the optimality of the Ingham-Karamata theorem

We study approximation properties of the Fréchet space of all continuously differentiable functions $τ$ such that $τ'(x)=o(1)$ and such that their Laplace transforms admit entire extensions to $\mathbb{C}$. As an application, these approximation results are combined with the open mapping theorem to show the optimality theorem for the Ingham-Karamata Tauberian theorem.

math.CA↗