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").
Categories: math.FA