Von Neumann's inequality explained

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."^[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\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 =inf ty it is true by straightforward calculation. S.W. Drury has shown in 2011 that the conjecture fails in the general case.^[2]

References

1. Web site: Department of Mathematics, Vanderbilt University Colloquium, AY 2007-2008 . 2008-03-11 . https://web.archive.org/web/20080316073544/http://www.math.vanderbilt.edu/~colloq/ . 2008-03-16 . dead .
2. A counterexample to a conjecture of Matsaev . 10.1016/j.laa.2011.01.022 . 2011 . Drury . S.W. . Linear Algebra and Its Applications . 435 . 2 . 323–329 .

See also

• Crouzeix's conjecture