On von Neumann's inequality on the polydisc

Abstract: 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 And^o show that if $d=1$ or $d=2$, the upper bound $|p(T)| \le |p|\infty$, holds, where the supremum norm is taken over the polydisc $\mathbb{D}^d$. We show that for $d=3$, there exists a universal constant $C$ such that $|p(T)| \le C |p|\infty$ for every homogeneous polynomial $p$. We also show that for general $d$ and arbitrary polynomials, the norm $|p(T)|$ is dominated by a certain Besov-type norm of $p$.