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 Andô show that if $d=1$ or $d=2$, the upper bound $\Vert p(T)\Vert \le \Vert p{\Vert }_{\infty }$, 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 $\Vert p(T)\Vert \le C\Vert p{\Vert }_{\infty }$ for every homogeneous polynomial $p$. We also show that for general $d$ and arbitrary polynomials, the norm $\Vert p(T)\Vert$ is dominated by a certain Besov-type norm of $p$.

28 pages; small changes