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 \( L^{p} \)
\( {\displaystyle ||P(T)||_{L^{p}\to L^{p}}\leq ||P(S)||_{\ell ^{p}\to \ell ^{p}}} \)
where S is the right-shift operator. The von Neumann inequality proves it true for p=2 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.[2]
References
"Department of Mathematics, Vanderbilt University Colloquium, AY 2007-2008". Archived from the original on 2008-03-16. Retrieved 2008-03-11.
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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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