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CONTRACTIONS ET HYPERDISTRIBUTIONS À SPECTRE DE CARLESON

Published online by Cambridge University Press:  01 August 1998

K. KELLAY
K. KELLAY
Affiliation:
UFR de Mathématiques et Informatique, Université de Bordeaux, 351 cours de la Libération, 33405 Talence Cedex, France. E-mail: [email protected] Current address: Département de Mathématiques et de Statistique, Université Laval, Québec, Canada G1K 7P4. E-mail: [email protected]
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Abstract

Let ω=(ωn)n[ges ]1 be a log concave sequence such that lim infn→+∞ ωn/nc>0 for some c>0 and ((log ωn)/nα)n[ges ]1 is nonincreasing for some α<1/2. We show that, if T is a contraction on the Hilbert space with spectrum a Carleson set, and if ∥T−n∥=O(ωn) as n tends to +∞ with [sum ]n[ges ]11/(n log ωn)=+∞, then T is unitary. On the other hand, if [sum ]n[ges ]11/(n log ωn)<+∞, then there exists a (non-unitary) contraction T on the Hilbert space such that the spectrum of T is a Carleson set, ∥T−n∥=O(ωn) as n tends to +∞, and lim supn→+∞ ∥T−n∥=+∞.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 58 , Issue 1 , August 1998 , pp. 185 - 196
Copyright
The London Mathematical Society 1998

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