IUMJ

Title: On the growth of powers of operators with spectrum contained in Cantor sets

Authors: Cyril Agrafeuil

Issue: Volume 54 (2005), Issue 5, 1473-1482

Abstract:

For $\xi \in (0,\frac{1}{2})$, we denote by $E_{\xi}$ the perfect symmetric set associated to $\xi$, that is, $$E_{\xi} = \{\exp (2i\pi (1-\xi) \sum_{n=1}^{+\infty} \epsilon_{n} \xi^{n-1}) \epsilon_{n} = 0 \mbox{ or } 1\ (n \geq 1)\}.$$ Let $s$ be a nonnegative real number, and $T$ be an invertible bounded operator on a Banach space with spectrum included in $E_{\xi}$. We show that if \begin{align*}\|T^{n}\| &= O(n^{s}),\quad n \to +\infty,\\ \|T^{-n}\| &= O(e^{n^{\beta}}), \quad n \to +\infty\end{align*} for some $\beta < \frac{\log(1/\xi) - \log2}{2log(1/\xi) - \log2}$, then for every $\varepsilon > 0$, $T$ satisfies the stronger property $$\|T^{-n}\| = O(n^{s+1/2+\varepsilon}), \quad n \to +\infty.$$ This result is a particular case of a more general result concerning operators with spectrum satisfying some geometrical conditions.