von Neumann's inequality for noncommuting contractions (Q2463619)

Let \(T_1, \dots, T_d\) be linear contractions on a complex Hilbert space and \(p\) be a complex polynomial in \(d\) variables which is a sum of \(d\) single variable polynomials. The operators \(T_1, \dots, T_d\) are not assumed to commute. It is shown that \[ \| p(T_1, \dots, T_d)\| \leq d \sin(\pi/(2d)) \sup_{| z_1|,\dots,| z_d|\leq 1} | p(z_1, \dots,z_d) | \] and that \(d \sin(\pi/(2d))\) is the best possible constant.