Von Neumann's inequality

Von Neumann's inequality

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." [ [http://www.math.vanderbilt.edu/~colloq/ Department of Mathematics, Vanderbilt University Colloquium, AY 2007-2008] ] 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.

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