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

$\backslash pi(X)\; =\; \backslash operatorname\{id\}\_H\; \backslash quad$

and for every ξ, η ∈ H, the set-function

$\backslash operatorname\{S\}\_\backslash pi(\backslash xi,\; \backslash eta)(A)\; =\; \backslash langle\; \backslash pi(A)\backslash xi\; \backslash mid\; \backslash eta\; \backslash rangle$

is a complex measure on M (that is, a complex-valued countably additive function).

If π is a projection-valued measure and

$A\; \backslash cap\; B\; =\; \backslash emptyset,$

then π(A), π(B) are orthogonal projections. From this follows that in general,

$\backslash pi(A)\; \backslash pi(B)\; =\; \backslash pi(A\; \backslash cap\; B).$

Example. Suppose (X, M, μ) is a measure space. Let π(A) be the operator of multiplication by the indicator function 1_A on L²(X). π is a projection-valued measure.

Extensions of projection-valued measures

If π is an additive projection-valued measure on (X, M), then the map

$\backslash mathbf\{1\}\_A\; \backslash mapsto\; \backslash pi(A)$

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 T_π(f) such that

$\backslash langle\; \backslash operatorname\{T\}\_\backslash pi(f)\; \backslash xi\; \backslash mid\; \backslash eta\; \backslash rangle\; =\; \backslash int\_X\; f(x)\; d\; \backslash operatorname\{S\}\_\backslash pi\; (\backslash xi,\backslash eta)(x)$

for all ξ, η ∈ H. The map

$f\; \backslash mapsto\; \backslash operatorname\{T\}\_\backslash pi(f)$

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 {H_x}_{x ∈ X} be a μ-measurable family of Hilbert spaces. For every A ∈ M, let π(A) be the operator of multiplication by 1_A on the Hilbert space

$\backslash int\_X^\backslash oplus\; H\_x\; \backslash \; d\; \backslash mu(x).$

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 if and only if there is a unitary operator U:H → K such that

$\backslash pi(A)\; =\; U^*\; \backslash rho(A)\; U\; \backslash quad$

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 {H_x}_{x ∈ X} , such that π is unitarily equivalent to multiplication by 1_A on the Hilbert space

$\backslash int\_X^\backslash oplus\; H\_x\; \backslash \; d\; \backslash mu(x).$

The measure class of μ and the measure equivalence class of the multiplicity function x → dim H_x completely characterize the projection-valued measure up to unitary equivalence.

A projection-valued measure π is homogeneous of multiplicity n if and only if the multiplicity function has constant value n. Clearly,

Theorem. Any projection-valued measure π is an orthogonal direct sum of homogeneous projection-valued measures:

$\backslash pi\; =\; \backslash bigoplus\_\{1\; \backslash leq\; n\; \backslash leq\; \backslash omega\}\; (\backslash pi\; |\; H\_n)$

where

$H\_n\; =\; \backslash int\_\{X\_n\}^\backslash oplus\; H\_x\; \backslash \; d\; (\backslash mu\; |\; X\_n)\; (x)$

and

$X\_n\; =\; \backslash \{x\; \backslash in\; X:\; \backslash operatorname\{dim\}\; H\_x\; =\; n\backslash \}.$

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.

Algebra | Functional analysis | Measure theory