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

摘要

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, {\iti.e.} there does not exist an invertible operator $S:\ \ell_2\to\ell_2$ suchthat $\|{S^{-1}T_\eps S}\|\leq 1$. This answers negatively a questionattributed to Halmos after his well known 1970 paper (``Ten problems in Hilbertspace").