On powers of operators with spectrum in cantor sets and spectral synthesis

Abstract

For $\xi \in (0,\frac{1}{2})$, let $E_{\xi }$ be the perfect symmetric set associated with $\xi$, that is $$E_{\xi }=\left\{\exp \right(2i\pi (1-\xi )\sum _{n=1}^{+\infty }{\epsilon }_n{\xi }^{n-1}):{\epsilon }_n=0\text{ or }1(n\ge 1)\}$$ and $$b(\xi )=\frac{\log \frac{1}{\xi }-\log 2}{2\log \frac{1}{\xi }-\log 2}.$$ Let $q\ge 3$ 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^β} \big), \, n \rightarrow +\infty \textrm{ for some } β< b(1/q),\end{eqnarray*} also satisfies the stronger property $\Vert T^{-n}\Vert =O\left(n^s\right),n\to +\infty .$ 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 $\omega (n)=(1+n{)}^s,(n\ge 0)$ and $C^{-1}(1+|n|{)}^s\le \omega (-n)\le C{e}^{n^{\beta }},(n\ge 0)$, 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 ${\sum }_{n=-\infty }^{+\infty }|\hat{f}(n)|\omega (n)<+\infty$.