For a contraction T acting on a Hilbert space and a polynomial p, then the norm of p(T) is bounded by the supremum of |p(z)| for z in the unit disk."^[1]

The inequality can be proved by considering the unitary dilation of T, for which the inequality is obvious.

This inequality is a specific case of Matsaev's conjecture. That is that for any polynomial P and contraction T on ${\displaystyle L^{p}}$ 

${\displaystyle ||P(T)||_{L^{p}\to L^{p}}\leq ||P(S)||_{\ell ^{p}\to \ell ^{p}}}$ 

where S is the right-shift operator. The von Neumann inequality proves it true for ${\displaystyle p=2}$  and for ${\displaystyle p=1}$  and ${\displaystyle p=\infty }$  it is true by straightforward calculation. S.W. Drury has shown in 2011 that the conjecture fails in the general case.^[2]

• Crouzeix's conjecture