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hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorZARRABI, Mohamed
dc.date.accessioned2024-04-04T03:09:58Z
dc.date.available2024-04-04T03:09:58Z
dc.date.created2017-06-09
dc.date.issued2018
dc.identifier.issn0022-247X
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/193652
dc.description.abstractEnFor $\xi \in \big( 0, \frac{1}{2} \big)$, 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.isoen
dc.publisherElsevier
dc.subject.enOperators
dc.subject.enGrowth of powers of operators
dc.subject.enSpectral synthesis
dc.subject.enCantor sets
dc.title.enOn powers of operators with spectrum in cantor sets and spectral synthesis
dc.typeArticle de revue
dc.subject.halMathématiques [math]/Analyse fonctionnelle [math.FA]
dc.identifier.arxiv1706.02943
bordeaux.journalJournal of Mathematical Analysis and Applications
bordeaux.page764-776
bordeaux.volume462
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.issue1
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.peerReviewedoui
hal.identifierhal-01535769
hal.version1
hal.popularnon
hal.audienceInternationale
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-01535769v1
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