# Von Neumann's inequality

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.

## 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."[1]

## 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 ${\displaystyle L^{p}}$

${\displaystyle ||P(T)||_{L^{p}\to L^{p}}\leq ||P(S)||_{\ell ^{p}\to \ell ^{p}}}$

where S is the right-shift operator. The von Neumann inequality proves it true for ${\displaystyle p=2}$ and for ${\displaystyle p=1}$ and ${\displaystyle p=\infty }$ it is true by straightforward calculation. S.W. Drury has shown in 2011 that the conjecture fails in the general case.[2]

## References

1. ^ "Department of Mathematics, Vanderbilt University Colloquium, AY 2007-2008". Archived from the original on 2008-03-16. Retrieved 2008-03-11.
2. ^ S.W. Drury, "A counterexample to a conjecture of Matsaev", Linear Algebra and its Applications, Volume 435, Issue 2, 15 July 2011, Pages 323-329