A polynomially bounded operator on Hilbert space which is not similar to a contraction
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Let > 0. We prove that there exists an operator T : l2 l2 such that for any polynomial P we have ||P(T)|| (1 + )||P||, but T is not similar to a contraction, i.e. there does not exist an invertible operator S : l2 l2 such that \S-1TS|| 1. This answers negatively a question attributed to Halmos after his well-known 1970 paper ("Ten problems in Hilbert space"). We also give some related finite-dimensional estimates.