A polynomially bounded operator on Hilbert space which is not similar to a contraction

Let E > 0. We prove that there exists an operator TE e2 "2 such that for any polynomial P we have IIP(TE)II not similar to a contraction, i.e. there does not exist an invertible operator S: f2 -e f2 such that JIS-1TESII tributed to Halmos after his well-known 1970 paper ("Ten problems in Hilbert space"). We also give some related finite-dimensional estimates. DEPARTMENT OF MATHEMATICS, TEXAS A&M UNIVERSITY, COLLEGE STATION, TEXAS 77843 UNIVERSITE PARIS VI, EQUIPE D7ANALYSE, CASE 186, 75252 PARIS CEDEX 05, FRANCE E-mail address: gipQccr. jussieu. fr This content downloaded from 157.55.39.104 on Sun, 19 Jun 2016 06:52:47 UTC All use subject to http://about.jstor.org/terms

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