In operator theory, von Neumann's inequality, due to John von Neumann, states that, for a fixed contraction T, the polynomial functional calculus map is itself a contraction.
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.
This inequality is a specific case of Matsaev's conjecture.
The von Neumann inequality proves it true for
Drury has shown in 2011 that the conjecture fails in the general case.