Von Neumann’s inequality for noncommuting contractions

Let T1,…,TdT1,…,Td be linear contractions on a complex Hilbert space and p a complex polynomial in d variables which is a sum of d   single variable polynomials. We show that the operator norm of p(T1,…,Td)p(T1,…,Td) is bounded bydsinπ2dsup|z1|,…,|zd|⩽1|p(z1,…,zd)|and that this is the best possible inequality of this type.