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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
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[1] DOI: 10.1090/S0273-0979-1994-00491-4 · Zbl 0803.46052
[2] DOI: 10.1016/0022-1236(90)90014-C · Zbl 0723.47013
[3] Esterle, J. f?r Reine Ang. Math. 450 pp 43– (1994)
[4] DOI: 10.1007/BF02392120 · Zbl 0449.47007
[5] Kahane, Ensembles parfaits et s?ries trigonom?triques (1963)
[6] Zarrabi, Ann. Inst. Fourier 43 pp 251– (1993) · Zbl 0766.47002
[7] DOI: 10.1307/mmj/1028999088 · Zbl 0133.37303
[8] Rudin, Real and complex analysis (1966)
[9] Nagy, Analyse harmonique des op?rateurs de l’espace de Hilbert (1967) · Zbl 0157.43201
[10] Zarrabi, Bull. Soc. Math. France 118 pp 241– (1990)
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