On von Neumann’s inequality on the polydisc

Given a d-tuple T of commuting contractions on Hilbert space and a polynomial p in d-variables, we seek upper bounds for the norm of the operator p(T). Results of von Neumann and Andô show that if \(d=1\) $d=1$ or \(d=2\) $d=2$ , the upper bound \(\Vert p(T)\Vert \le \Vert p\Vert _\infty \) $\Vert p(T)\Vert \le {\Vert p\Vert }_{\infty }$ , holds, where the supremum norm is taken over the polydisc \(\mathbb {D}^d\) ${\mathbb{D}}^d$ . We show that for \(d=3\) $d=3$ , there exists a universal constant C such that \(\Vert p(T)\Vert \le C \Vert p\Vert _\infty \) $\Vert p(T)\Vert \le {C\Vert p\Vert }_{\infty }$ for every homogeneous polynomial p. We also show that for general d and arbitrary polynomials, the norm \(\Vert p(T)\Vert \) $\Vert p(T)\Vert$ is dominated by a certain Besov-type norm of p.