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.

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."{{Cite web |url=http://www.math.vanderbilt.edu/~colloq/ |title=Department of Mathematics, Vanderbilt University Colloquium, AY 2007-2008 |access-date=2008-03-11 |archive-url=https://web.archive.org/web/20080316073544/http://www.math.vanderbilt.edu/~colloq/ |archive-date=2008-03-16 |url-status=dead }}

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 $L^p$

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

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 shown in 2011 that the conjecture fails in the general case.{{cite journal | url=http://www.sciencedirect.com/science/article/pii/S0024379511000589 | doi=10.1016/j.laa.2011.01.022 | title=A counterexample to a conjecture of Matsaev | date=2011 | last1=Drury | first1=S.W. | journal=Linear Algebra and Its Applications | volume=435 | issue=2 | pages=323–329 }}

Von Neumann's inequality

Statement

Proof

Generalizations

References

See also