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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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


Author: Gilles Pisier
Journal: J. Amer. Math. Soc. 10 (1997), 351-369
MSC (1991): Primary 47A20, 47B35, 47D25, 47B47; Secondary 47A56, 42B30
MathSciNet review: 1415321
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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
Email: gip@ccr.jussieu.fr

DOI: http://dx.doi.org/10.1090/S0894-0347-97-00227-0
PII: S 0894-0347(97)00227-0
Received by editor(s): March 11, 1996
Received by editor(s) in revised form: October 11, 1996
Article copyright: © Copyright 1997 American Mathematical Society