On powers of operators with spectrum in cantor sets and spectral synthesis
| hal.structure.identifier | Institut de Mathématiques de Bordeaux [IMB] | |
| dc.contributor.author | ZARRABI, Mohamed | |
| dc.date.accessioned | 2024-04-04T03:09:58Z | |
| dc.date.available | 2024-04-04T03:09:58Z | |
| dc.date.created | 2017-06-09 | |
| dc.date.issued | 2018 | |
| dc.identifier.issn | 0022-247X | |
| dc.identifier.uri | https://oskar-bordeaux.fr/handle/20.500.12278/193652 | |
| dc.description.abstractEn | For $\xi \in \big( 0, \frac{1}{2} \big)$, let $E_{\xi}$ be the perfect symmetric set associated with $\xi$, that is $$ E_{\xi} = \Big\{ \exp \Big( 2i \pi (1-\xi) \sum_{n = 1}^{+\infty} \epsilon_{n} \xi^{n-1} \Big) : \, \epsilon_{n} = 0 \textrm{ or } 1 \quad (n \geq 1) \Big\} $$ and $$ b(\xi) = \frac{\log{\frac{1}{\xi}} - \log{2}}{2\log{\frac{1}{\xi}} - \log{2}}. $$ Let $q\geq 3$ be an integer and $s$ be a nonnegative real number. We show that any invertible operator $T$ on a Banach space with spectrum contained in $E_{1/q}$ that satisfies \begin{eqnarray*} & & \big\| T^{n} \big\| = O \big( n^{s} \big), \,n \rightarrow +\infty \\ & \textrm{and} & \big\| T^{-n} \big\| = O \big( e^{n^{\beta}} \big), \, n \rightarrow +\infty \textrm{ for some } \beta < b(1/q), \end{eqnarray*} also satisfies the stronger property $\big\| T^{-n} \big\| = O \big( n^{s} \big), \, n \rightarrow +\infty.$ We also show that this result is false for $E_\xi$ when $1/\xi$ is not a Pisot number and that the constant $b(1/q)$ is sharp. As a consequence we prove that, if $\omega$ is a submulticative weight such that $\omega(n)=(1+n)^s, \, (n \geq 0)$ and $C^{-1} (1+|n|)^s \leq \omega(-n) \leq C e^{n^{\beta}},\, (n\geq 0)$, for some constants $C>0$ and $\beta < b( 1/q),$ then $E_{1/q}$ satisfies spectral synthesis in the Beurling algebra of all continuous functions $f$ on the unit circle $\mathbb{T}$ such that $\sum_{n = -\infty}^{+\infty} | \widehat{f}(n) | \omega (n) < +\infty$. | |
| dc.language.iso | en | |
| dc.publisher | Elsevier | |
| dc.subject.en | Operators | |
| dc.subject.en | Growth of powers of operators | |
| dc.subject.en | Spectral synthesis | |
| dc.subject.en | Cantor sets | |
| dc.title.en | On powers of operators with spectrum in cantor sets and spectral synthesis | |
| dc.type | Article de revue | |
| dc.subject.hal | Mathématiques [math]/Analyse fonctionnelle [math.FA] | |
| dc.identifier.arxiv | 1706.02943 | |
| bordeaux.journal | Journal of Mathematical Analysis and Applications | |
| bordeaux.page | 764-776 | |
| bordeaux.volume | 462 | |
| bordeaux.hal.laboratories | Institut de Mathématiques de Bordeaux (IMB) - UMR 5251 | * |
| bordeaux.issue | 1 | |
| bordeaux.institution | Université de Bordeaux | |
| bordeaux.institution | Bordeaux INP | |
| bordeaux.institution | CNRS | |
| bordeaux.peerReviewed | oui | |
| hal.identifier | hal-01535769 | |
| hal.version | 1 | |
| hal.popular | non | |
| hal.audience | Internationale | |
| hal.origin.link | https://hal.archives-ouvertes.fr//hal-01535769v1 | |
| bordeaux.COinS | ctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Journal%20of%20Mathematical%20Analysis%20and%20Applications&rft.date=2018&rft.volume=462&rft.issue=1&rft.spage=764-776&rft.epage=764-776&rft.eissn=0022-247X&rft.issn=0022-247X&rft.au=ZARRABI,%20Mohamed&rft.genre=article |
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