Projection-valued Measure

In mathematics, projection-valued measures are used to express results in spectral theory. A projection-valued measure on a measurable space (X, M) is a mapping π from M to the set of self-adjoint projections on a Hilbert space H such that
\pi(X) = 1_H \quad
and for every ξ η ∈ H, the set-function
\operatorname{S}_\pi(\xi, \eta)(A) = \langle \pi(A)\xi \mid \eta \rangle
is a complex-valued countably additive measure on M. If π is a projection-valued measure and
A \cap B = \emptyset,
then π(A), π(B) are orthogonal projections. From this follows that in general,
\pi(A) \pi(B) = \pi(A \cap B).
Example. Suppose (X, M, μ) is a measure space. Let π(A) be the operator of multiplication by 1A on L2(X). π is a projection-valued measure.

Extensions of projection-valued measures

If π is a additive projection-valued measure on (X, M), then the map
\mathbf{1}_A \mapsto \pi(A)
extends to a linear map on the vector space of step functions on X. In fact, it is easy to check that this map is a ring homomorphism. In fact this map extends in a canonical way to all bounded complex-valued Borel functions on X. Theorem. For any bounded M-measurable function f on X, there is a unique bounded linear operator Tπ(f) such that
\langle \operatorname{T}_\pi(f) \xi \mid \eta \rangle = \int_X f(x) d \operatorname{S}_\pi (\xi,\eta)(x)
for all ξ, η ∈ H. The map
f \mapsto \operatorname{T}_\pi(f)
is a homomorphism of rings.

Structure of projection-valued measures

First we provide a general example of projection-valued measure based on direct integrals. Suppose (X, M, μ) is a measure space and let {Hx}x ∈ X be a μ-measurable family of Hilbert spaces. For every A ∈ M, let π(A) be the operator of multiplication by 1A on the Hilbert space
\int_X^\oplus H_x \ d \mu(x).
Then π is a projection-valued measure on (X, M). Suppose π ρ are projection-valued measures on (X, M) with values in the projections of H, K. π, ρ are unitarily equivalent iff there is a unitary operator U:H → K such that
\pi(A) = U^* \rho(A) U \quad
for every A ∈ M. Theorem. If (X, M) is a standard Borel space, then for every projection-valued measure π on (X, M), there is a Borel measure μ and a μ-measurable family of Hilbert spaces {Hx}x ∈ X , such that π is unitarily equivalent to multiplication by 1A on the Hilbert space
\int_X^\oplus H_x \ d \mu(x).
The measure class of μ and the measure equivalence class of the multiplicity function x → dim Hx completely characterize the projection-valued measure up to unitary equivalence. A projection-valued measure π is homogeneous of multiplicity n iff the multiplicity function has constant value n. Clearly, Theorem. Any projection-valued measure π is an orthogonal direct sum of homogeneous projection-valued measures:
\pi = \bigoplus_{1 \leq n \leq \omega} (\pi | H_n)
where
H_n = \int_{X_n}^\oplus H_x \ d (\mu | X_n) (x)
and
X_n = \{x \in X: \operatorname{dim} H_x = n\}.

References

  • G. W. Mackey, The Theory of Unitary Group Representations, The University of Chicago Press, 1976
  • V. S. Varadarajan, Geometry of Quantum Theory V2, Springer Verlag, 1970.

 

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