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.
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)
\{Bi\}
\langleE(\cupiBi)x,y\rangle=\sumi\langleE(Bi)x,y\rangle
for all x and y. Some terminology for describing such measures are:
B → \langleE(B)x,y\rangle
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.
|E|=\supB\|E(B)\|<infty
E(B1\capB2)=E(B1)E(B2)
B1,B2
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
\PhiE:C(X) → L(H)
\langle\PhiE(f)h1,h2\rangle=\intXf(x)\langleE(dx)h1,h2\rangle
The boundedness of E implies, for all h of unit norm
\langle\PhiE(f)h,h\rangle=\intXf(x)\langleE(dx)h,h\rangle\leq\|f\|infty ⋅ |E|.
This shows
\PhiE(f)
\PhiE
The properties of
\PhiE
\PhiE
\PhiE
h1,h2\inH
\langle\PhiE(fg)h1,h2\rangle=\intXf(x) ⋅ g(x) \langleE(dx)h1,h2\rangle=\langle\PhiE(f)\PhiE(g)h1,h2\rangle.
Take f and g to be indicator functions of Borel sets and we see that
\PhiE
\PhiE
\langle\PhiE({\barf})h1,h2\rangle=\langle\PhiE(f)*h1,h2\rangle.
The LHS is
\intX{\barf} \langleE(dx)h1,h2\rangle,
and the RHS is
\langleh1,\PhiE(f)h2\rangle=\overline{\langle\PhiE(f)h2,h1\rangle}=\intX{\barf}(x) \overline{\langleE(dx)h2,h1\rangle}=\intX{\barf}(x) \langleh1,E(dx)h2\rangle
So, taking f a sequence of continuous functions increasing to the indicator function of B, we get
\langleE(B)h1,h2\rangle=\langleh1,E(B)h2\rangle
\PhiE
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
V:K → H
E(B)=VF(B)V*.
We now sketch the proof. The argument passes E to the induced map
\PhiE
\PhiE
\PhiE
\PhiE
\pi:C(X) → L(K)
V:K → H
\PhiE(f)=V\pi(f)V*.
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.
In the finite-dimensional case, there is a somewhat more explicit formulation.
Suppose now
X=\{1,...c,n\}
Cn
Ei
Of particular interest is the special case when
\sumiEi=I
\PhiE
Ei
xix
* | |
i |
xi\inCm
n<m
n=m
n | |
\sum | |
i=1 |
xi
* | |
x | |
i |
=I
\{xi\}
n>m
\{Ei\}
M=\begin{bmatrix}x1 … xn\end{bmatrix}
MM*=I
(n-m) x n
U=\begin{bmatrix}M\ N\end{bmatrix}
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.