Polynomial bounds for small matrices
Résumé
The polynomial bound of a Hilbert space operator $T$ is the quantity $K(T) = \sup\{\| p(T)\|\ p \ \hbox{\rm is \ a \ polynomial\ mapping \ the \ unit \ disc \ into\ itself} \}$. One of the Halmos "Ten problems" asked whether $K(T) < \infty$ implies that $T$ is similar to a contraction. This question was settled in the negative by Pisier after many years. The finite-dimensional version of the problem asks to what extent $K(T)$ can be exceeded by $M(T) = \infty\{\|S\|\|S^{-1}\| STS^{-1}$ is a contraction$\}$. We establish some general criteria for the equality $K(T) = M(T)$ and show that $K(T) < M(T)$ can occur even $ 3 x 3$ for matrices.