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On contractions with spectrum contained in the Cantor set. (English) Zbl 0830.47001
Summary: For $$\xi\in (0, {1\over 2})$$ let $$E_\xi$$ be the perfect symmetric set of constant ratio $$\xi$$ and set $b(\xi)= {\log 1/\xi- \log 2\over 2\log 1/\xi- \log 2}.$ It was shown by the first author that if $$T$$ is a contraction on the Hilbert space $$H$$ with spectrum contained in $$E_\xi$$, and if $$\log|T^{- n}|= O(n^\alpha)$$ as $$n\to \infty$$ for some $$\alpha< b(\xi)$$, then $$T$$ is unitary. In the other direction, we show here that there exists a (non-unitary) contraction $$T$$ on $$H$$ such that $$Sp T= E_\xi$$, $$\log|T^{- n}|= O(n^{b(\xi)})$$ as $$n\to \infty$$, and $$\limsup_{n\to \infty}|T^{- n}|= \infty$$.

##### MSC:
 47A10 Spectrum, resolvent
##### Keywords:
spectrum contained in the Cantor set; contraction
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##### References:
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