Esscher principle: Difference between revisions
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In [[operator theory]], '''von Neumann's inequality''', due to [[John von Neumann]], states that, for a [[Contraction (operator theory)|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]]."<ref>[http://www.math.vanderbilt.edu/~colloq/ Department of Mathematics, Vanderbilt University Colloquium, AY 2007-2008]</ref> In other words, for a fixed contraction ''T'', the [[polynomial functional calculus]] map is itself a contraction. The inequality can be proved by considering the [[unitary dilation]] of ''T'', for which the inequality is obvious. | |||
This inequality is a specific case of Matsaev's conjecture. That is that for any polynomial ''P'' and contraction ''T'' on <math>L^p</math> | |||
:<math>||P(T)||_{L^p} \le ||P(S)||_{\ell^p}</math> | |||
where ''S'' is the right-shift operator. The von Neumann inequality proves it true for <math>p=2</math> and for <math>p=1</math> and <math>p=\infty</math> it is true by straightforward calculation. | |||
S.W. Drury has recently shown that the conjecture fails in the general case.<ref>[http://www.sciencedirect.com/science/article/pii/S0024379511000589 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 ]</ref> | |||
==References== | |||
<references/> | |||
[[Category:Operator theory]] | |||
[[Category:Inequalities]] | |||
{{mathanalysis-stub}} | |||
Latest revision as of 11:58, 23 January 2014
In operator theory, von Neumann's inequality, due to John von Neumann, states that, 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] In other words, for a fixed contraction T, the polynomial functional calculus map is itself a contraction. The inequality can be proved by considering the unitary dilation of T, for which the inequality is obvious.
This inequality is a specific case of Matsaev's conjecture. That is that for any polynomial P and contraction T on
where S is the right-shift operator. The von Neumann inequality proves it true for and for and it is true by straightforward calculation. S.W. Drury has recently shown that the conjecture fails in the general case.[2]