In operator theory, Naimark's dilation theorem is a result that characterizes positive operator valued measures. It can be viewed as a consequence of Stinespring's dilation theorem.
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Note
In the mathematical literature, one may also find other results that bear Naimark's name.
Spelling
In the physics literature, it is common to see the spelling "Neumark" instead of "Naimark." This is due to translating between the Russian alphabet spelling and the spelling in European languages (namely English and German) using the Roman alphabet.
Some preliminary notions
Let X be a compact Hausdorff space, H be a Hilbert space, and L(H) the Banach space of bounded operators on H. A mapping E from the Borel σ-algebra on X to
for all x and y. Some terminology for describing such measures are:
is a regular Borel measure, meaning all compact sets have finite total variation and the measure of a set can be approximated by those of open sets.
We will assume throughout that E is regular.
Let C(X) denote the abelian C*-algebra of continuous functions on X. If E is regular and bounded, it induces a map
The boundedness of E implies, for all h of unit norm
This shows
The properties of
Take f and g to be indicator functions of Borel sets and we see that
The LHS is
and the RHS is
So, taking f a sequence of continuous functions increasing to the indicator function of B, we get
Naimark's theorem
The theorem reads as follows: Let E be a positive L(H)-valued measure on X. There exists a Hilbert space K, a bounded operator
Proof
We now sketch the proof. The argument passes E to the induced map
Since π is a *-homomorphism, its corresponding operator-valued measure F is spectral and self adjoint. It is easily seen that F has the desired properties.
Finite-dimensional case
In the finite-dimensional case, there is a somewhat more explicit formulation.
Suppose now
Of particular interest is the special case when
-
n = m and E is already a projection-valued measure (because∑ i = 1 n x i x i ∗ = I if and only if{ x i } is an orthonormal basis), -
n > m and{ E i } does not consist of mutually orthogonal projections.
For the second possibility, the problem of finding a suitable projection-valued measure now becomes the following problem. By assumption, the non-square matrix
is an isometry, that is
is a n × n unitary matrix, the projection-valued measure whose elements are projections onto the column vectors of U will then have the desired properties. In principle, such a N can always be found.