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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 \(\limsup_{n\to \infty}|T^{- n}|= \infty\).

47A10 Spectrum, resolvent
Full Text: DOI
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