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

Pisier, Gilles^1, 2

¹ Department of Mathematics, Texas A&M University, College Station, Texas 77843
² Université Paris VI, Equipe d’Analyse, Case 186, 75252 Paris Cedex 05, France

Journal of the American Mathematical Society, Tome 10 (1997) no. 2, pp. 351-369

Résumé

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.

Détail

DOI : 10.1090/S0894-0347-97-00227-0

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