On von Neumann’s inequality on the polydisc

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AbstractGiven 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 $$\Vert p(T)\Vert \le \Vert p\Vert _\infty $$ ‖ p ( T ) ‖ ≤ ‖ p ‖ ∞ , holds, where the supremum norm is taken over the polydisc $$\mathbb {D}^d$$ D d . We show that for $$d=3$$ d = 3 , there exists a universal constant C such that $$\Vert p(T)\Vert \le C \Vert p\Vert _\infty $$ ‖ p ( T ) ‖ ≤ C ‖ p ‖ ∞ for every homogeneous polynomial p. We also show that for general d and arbitrary polynomials, the norm $$\Vert p(T)\Vert $$ ‖ p ( T ) ‖ is dominated by a certain Besov-type norm of p.