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
where S is the right-shift operator. The von Neumann inequality proves it true for and for and it is true by straightforward calculation. S.W. Drury has shown in 2011 that the conjecture fails in the general case.