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.
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 L p {\displaystyle L^{p}}
| | P ( T ) | | L p → L p ≤ | | P ( S ) | | ℓ p → ℓ p {\displaystyle ||P(T)||_{L^{p}\to L^{p}}\leq ||P(S)||_{\ell ^{p}\to \ell ^{p}}}
where S is the right-shift operator. The von Neumann inequality proves it true for p = 2 {\displaystyle p=2} and for p = 1 {\displaystyle p=1} and p = ∞ {\displaystyle p=\infty } it is true by straightforward calculation. S.W. Drury has shown in 2011 that the conjecture fails in the general case.
References
See also Crouzeix's conjecture