Spectral characterizations of stable operator semigroups

Abstract

We introduce the notion of local pseudofunction spectrum σ_PF(A) for the infinitesimal generator A of a bounded C₀-semigroup 𝒯 = (T(t))_{t ≥ 0} on a Banach space X and show it is the right spectral concept to deliver a full characterization of the strong stability of 𝒯: ∀ x ∈ X : lim_{t → ∞} | T(t) x |_X = 0 Longleftrightarrow σ_PF(A) = ∅. 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 C₀-semigroups with countable local pseudofunction spectrum. In addition, we prove optimal Tauberian theorems of Katznelson-Tzafriri type and discuss connections with Wiener kernels.

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BibTeX RIS

BibTeX

@misc{callewaert2026spectral,
  title = {{Spectral characterizations of stable operator semigroups}},
  author = {Morgan Callewaert and Lenny Neyt and Jasson Vindas},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.15926},
  eprint = {2606.15926},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.15926},
}

RIS

TY  - UNPB
TI  - Spectral characterizations of stable operator semigroups
AU  - Morgan Callewaert
AU  - Lenny Neyt
AU  - Jasson Vindas
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2606.15926
UR  - https://arxiv.org/abs/2606.15926
AB  - We introduce the notion of local pseudofunction spectrum σ_PF(A) for the infinitesimal generator A of a bounded C_0-semigroup 𝒯 = (T(t))_t ≥ 0 on a Banach space X and show it is the right spectral concept to deliver a full characterization of the strong stability of 𝒯: ∀ x ∈ X : lim_t → ∞ | T(t) x |_X = 0 Longleftrightarrow σ_PF(A) = ∅. 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.
ER  -

Abstract

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