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

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**10**(1997), 351-369 Request permission

## Abstract:

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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## Additional 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: gip@ccr.jussieu.fr
- 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: https://doi.org/10.1090/S0894-0347-97-00227-0
- MathSciNet review: 1415321