A Polynomially Bounded Operator on Hilbert Space Which Is Not Similar to a Contraction

Let ε > 0. We prove that there exists an operator Tε : `2 → `2such that for any polynomial P we have ‖P (Tε)‖ ≤ (1 +ε)‖P‖∞, but Tε isnot similar to a contraction, i.e. there does not exist an invertible operatorS : `2 → `2 such that‖S−1TεS‖ ≤ 1. This answers negatively a question at-tributed to Halmos after his well-known 1970 paper (“Ten problems in Hilbertspace”). We also give some related finite-dimensional estimates. Department of Mathematics, Texas A&M; University, College Station, Texas 77843 Université Paris VI, Equipe d’Analyse, Case 186, 75252 Paris Cedex 05, FranceE-mail address: [email protected] License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use