High Quality Content by WIKIPEDIA articles! In operator theory, von Neumann's inequality, due to John von Neumann, states that, 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." In other words, for a fixed contraction T, the polynomial functional calculus map is itself a contraction. The inequality can be proved by considering the unitary dilation of T, for which the inequality is obvious. The von Neumann inequality proves it true for p = 2 and for p = 1 and p=infinityit is true by straightforward calculation. The other cases are open questions.

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