Von Neumann's Theorem
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, von Neumann's theorem is a result in the
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 operato ...
of
linear operator In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s on
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
s.


Statement of the theorem

Let G and H be Hilbert spaces, and let T : \operatorname(T) \subseteq G \to H be an
unbounded operator In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases. The t ...
from G into H. Suppose that T is a closed operator and that T is densely defined, that is, \operatorname(T) is
dense Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...
in G. Let T^* : \operatorname\left(T^*\right) \subseteq H \to G denote the
adjoint In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...
of T. Then T^* T is also densely defined, and it is
self-adjoint In mathematics, an element of a *-algebra is called self-adjoint if it is the same as its adjoint (i.e. a = a^*). Definition Let \mathcal be a *-algebra. An element a \in \mathcal is called self-adjoint if The set of self-adjoint elements ...
. That is, \left(T^* T\right)^* = T^* T and the operators on the right- and left-hand sides have the same dense domain in G.


References

{{Functional analysis Operator theory Theorems in functional analysis