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

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

2006-01-25

arxiv.org/abs/math/0601609v1

Indiana Univ. Math. J. 54 (2005), no. 5, 1473--1481

Mathematics

Functional Analysis

9 pages

Scientific paper

For $\xi \in \big(0, {1/2} \big)$, we denote by $E_{\xi}$ the perfect symmetric set associated to $\xi$, that is $$ E_{\xi} = \Big\{\exp \big(2i \pi (1-\xi) \dsp \sum_{n = 1}^{+\infty} \epsilon_{n} \xi^{n-1} \big) : \epsilon_{n} = 0 \textrm{or} 1 \quad (n \geq 1) \Big\}. $$ 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{eqnarray*} & & \big\| T^{n} \big\| = O \big(n^{s} \big), n \to +\infty & \textrm{and} & \big\| T^{-n} \big\| = O \big(e^{n^{\beta}} \big), n \to +\infty \textrm{for some} \beta < \frac{\log{\frac{1}{\xi}} - \log{2}}{2\log{\frac{1}{\xi}} - \log{2}}, \end{eqnarray*} then for every $\e > 0$, $T$ satisfies the stronger property $$ \big\| T^{-n} \big\| = O \big(n^{s+{1/2}+\e} \big), n \to +\infty. $$ This result is a particular case of a more general result concerning operators with spectrum satisfying some geometrical conditions.

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