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.

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 L(H) is called an operator-valued measure if it is weakly countably additive, that is, for any disjoint sequence of Borel sets { B_i }, we have 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 in the obvious way: The boundedness of E implies, for all h of unit norm This shows is a bounded operator for all f, and \Phi _E itself is a bounded linear map as well. The properties of \Phi_E are directly related to those of E: Take f and g to be indicator functions of Borel sets and we see that \Phi _E is a homomorphism if and only if E is spectral. The LHS is and the RHS is So, taking f a sequence of continuous functions increasing to the indicator function of B, we get, i.e. E(B) is self adjoint.

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, and a self-adjoint, spectral L(K)-valued measure on X, F, such that

Proof

We now sketch the proof. The argument passes E to the induced map \Phi_E and uses Stinespring's dilation theorem. Since E is positive, so is \Phi_E as a map between C*-algebras, as explained above. Furthermore, because the domain of \Phi _E, C(X), is an abelian C*-algebra, we have that \Phi_E is completely positive. By Stinespring's result, there exists a Hilbert space K, a *-homomorphism, and operator such that 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, therefore C(X) is the finite-dimensional algebra , and H has finite dimension m. A positive operator-valued measure E then assigns each i a positive semidefinite m × m matrix E_i. Naimark's theorem now states that there is a projection-valued measure on X whose restriction is E. Of particular interest is the special case when where I is the identity operator. (See the article on POVM for relevant applications.) In this case, the induced map \Phi_E is unital. It can be assumed with no loss of generality that each E_i takes the form x_ix_i^* for some potentially subnorrmalized vector. Under such assumptions, the case n < m is excluded and we must have either 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 a co-isometry, that is M M^* = I. If we can find a matrix N where 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.

Spelling

In the physics literature, it is common to see the spelling “Neumark” instead of “Naimark.” The latter variant is according to the romanization of Russian used in translation of Soviet journals, with diacritics omitted (originally Naĭmark). The former is according to the etymology of the surname of Mark Naimark.