operator theory In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operat ...

, von Neumann's inequality, due to

John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

, states that, for a fixed contraction ''T'', the polynomial functional calculus map is itself a contraction.

Formal statement

For a

contraction Contraction may refer to: Linguistics * Contraction (grammar), a shortened word * Poetic contraction, omission of letters for poetic reasons * Elision, omission of sounds ** Syncope (phonology), omission of sounds in a word * Synalepha, merged ...

''T'' acting on a

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

and a polynomial ''p'', then the norm of ''p''(''T'') is bounded by the

supremum In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...

of , ''p''(''z''), for ''z'' in the

unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose di ...

Proof

The inequality can be proved by considering the

unitary dilation In operator theory, a dilation of an operator ''T'' on a Hilbert space ''H'' is an operator on a larger Hilbert space ''K'', whose restriction to ''H'' composed with the orthogonal projection onto ''H'' is ''T''. More formally, let ''T'' be a boun ...

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), , _ \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.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
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References

Operator theory Inequalities John von Neumann