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Journal of the London Mathematical Society 1998 58(1):185-196; doi:10.1112/S0024610798006309
© 1998 by London Mathematical Society
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© The London Mathematical Society

Contractions et Hyperdistributions à Spectre de Carleson

K. Kellay

UFR de Mathématiques et Informatique, Université de Bordeaux 351 cours de la Libération, 33405 Talence Cedex, France. E-mail: kellay{at}math.u-bordeaux.fr

Received 20 July 1995. Revision received 17 January 1996.

Let {omega}=({omega}n)n≥1 be a log concave sequence such that lim infn->+{infty}{omega}n/nc>0 for some c>0 and ((log {omega}n)/n{alpha})n≥1 is nonincreasing for some {alpha}<1/2. We show that, if T is a contraction on the Hilbert space with spectrum a Carleson set, and if ||T–n||=O({omega}n) as n tends to +{infty} with {sum}n≥11/(n log {omega}n)=+{infty}, then T is unitary. On the other hand, if {sum}n≥11/(n log {omega}n)<+{infty}, 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({omega}n) as n tends to +{infty}, and lim supn->+{infty}||T–n||=+{infty}.

Département de Mathématiques et de Statistique, Université Laval, Québec, Canada G1K 7P4. E-mail: kellay{at}mat.ulaval.ca


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