Optimal decay of semi-uniformly stable operator semigroups with empty spectrum
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917256032681984 |
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| author | Callewaert, Morgan Neyt, Lenny Vindas, Jasson |
| author_facet | Callewaert, Morgan Neyt, Lenny Vindas, Jasson |
| contents | 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 $x \in X$, $$ \limsup_{t\to\infty} \frac{\|T(t)A^{-1}x\|_{X}}{r(t)}=\infty. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20376 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal decay of semi-uniformly stable operator semigroups with empty spectrum Callewaert, Morgan Neyt, Lenny Vindas, Jasson Functional Analysis Analysis of PDEs Classical Analysis and ODEs Spectral Theory Primary 47D06, Secondary 34G10, 40E05, 44A10 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 $x \in X$, $$ \limsup_{t\to\infty} \frac{\|T(t)A^{-1}x\|_{X}}{r(t)}=\infty. $$ |
| title | Optimal decay of semi-uniformly stable operator semigroups with empty spectrum |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs Spectral Theory Primary 47D06, Secondary 34G10, 40E05, 44A10 |
| url | https://arxiv.org/abs/2507.20376 |