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

Source abstract

Let ε > 0 \varepsilon >0 . We prove that there exists an operator T ε : ℓ 2 → ℓ 2 T_{\varepsilon }:\ell _{2}\to \ell _{2} such that for any polynomial P P we have ‖ P ( T ε ) ‖ ≤ ( 1 + ε ) ‖ P ‖ ∞ \|{P(T_{\varepsilon })}\| \leq (1+\varepsilon ) \|{P}\|_{\infty } , but T ε T_{\varepsilon } is not similar to a contraction, i.e. there does not exist an invertible operator S : ℓ 2 → ℓ 2 S: \ell _{2}\to \ell _{2} such that ‖ S − 1 T ε S ‖ ≤ 1 \|{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.