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 $L^p$

$||P(T)||_{L^p} \le ||P(S)||_{\ell^p}$

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

References

1. Department of Mathematics, Vanderbilt University Colloquium, AY 2007-2008
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