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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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A polynomially bounded operator on Hilbert space which is not similar to a contraction
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by Gilles Pisier
J. Amer. Math. Soc. 10 (1997), 351-369


Let $\varepsilon >0$. We prove that there exists an operator $T_{\varepsilon }:\ell _{2}\to \ell _{2}$ such that for any polynomial $P$ we have $\|{P(T_{\varepsilon })}\| \leq (1+\varepsilon ) \|{P}\|_{\infty }$, but $T_{\varepsilon }$ is not similar to a contraction, i.e. there does not exist an invertible operator $S: \ell _{2}\to \ell _{2}$ such that $\|{S^{-1}T_{\varepsilon }S}\|\leq 1$. 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.
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Bibliographic Information
  • Gilles Pisier
  • Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843; Université Paris VI, Equipe d’Analyse, Case 186, 75252 Paris Cedex 05, France
  • MR Author ID: 140010
  • Email:
  • Received by editor(s): March 11, 1996
  • Received by editor(s) in revised form: October 11, 1996
  • © Copyright 1997 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 10 (1997), 351-369
  • MSC (1991): Primary 47A20, 47B35, 47D25, 47B47; Secondary 47A56, 42B30
  • DOI:
  • MathSciNet review: 1415321