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On contractions with spectrum contained in the Cantor set

Published online by Cambridge University Press:  24 October 2008

J. Esterle ,
M. Zarrabi  and
M. Rajoelina
J. Esterle
Affiliation:
U.F.R. de Mathématiques et Informatique, Université Bordeaux I, 351, Cours de la Libération, 33405 Talence
M. Zarrabi
Affiliation:
U.F.R. de Mathématiques et Informatique, Université Bordeaux I, 351, Cours de la Libération, 33405 Talence
M. Rajoelina
Affiliation:
Département de Mathématiques, Faculté des Sciences, B.P. 906 Antananarivo 101, Madagascar
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Abstract

For ξ ε (0, ½) let Eξ be the perfect symmetric set of constant ratio ξ and set

It was shown by the first author that if T is a contraction on the Hilbert space H with spectrum contained in Eξ, and if log ∥T−n∥ = O(nα) as n → ∞ for some α < b(ξ), then T is unitary. In the other direction, we show here that there exists a (non-unitary) contraction T on H such that SpT = Eξ, log ∥T-n∥ = O(nb(ξ)) as n → ∞, and lim supn → ∞ ∥T−n∥ = ∞.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 117 , Issue 2 , March 1995 , pp. 339 - 343
Copyright
Copyright © Cambridge Philosophical Society 1995

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References

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