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 p(T ) ≤ p∞, holds, where the supremum norm is taken over the polydisc Dd . We show that for d = 3, there exists a universal constant C such that p(T ) ≤ Cp∞ 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.