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.
Contents
Formal statement
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."
Proof
The inequality can be proved by considering the unitary dilation of T, for which the inequality is obvious.
Generalizations
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