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

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

1996-02-27

arxiv.org/abs/math/9602207v1

Journal Amer. Math. Soc. 10 (1997) 351-369.

Mathematics

Functional Analysis

Scientific paper

Let $\eps >0$. We prove that there exists an operator $T_\eps:\ell_2\to\ell_2$, such that for any polynomial $P$ we have $\|{P(T)}\| \leq(1+\eps)\|{P}\|_\infty$, but which is not similar to a contraction, {\it i.e.} there does not exist an invertible operator $S:\ \ell_2\to\ell_2$ such that $\|{S^{-1}T_\eps S}\|\leq 1$. This answers negatively a question attributed to Halmos after his well known 1970 paper (``Ten problems in Hilbert space").

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