Projection-valued measure

In mathematics, particularly functional analysis a projection-valued measure is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a Hilbert space. Projection-valued measures are used to express results in spectral theory, such as the spectral theorem for self-adjoint operators.

Formal definition

A projection-valued measure on a measurable space ("X", "M"), where "M" is a σ-algebra of subsets of "X", is a mapping π from "M" to the set of self-adjoint projections on a Hilbert space "H" such that

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

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

:$A\; mapsto\; langle\; pi(A)xi\; mid\; eta\; angle$

is a complex measure on "M" (that is, a complex-valued countably additive function). We denote this measure by $operatorname\{S\}\_pi(xi,\; eta)$.

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 the indicator function 1_"A" on "L"²("X"). Then 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

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

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 T_π("f") such that

: $langle\; operatorname\{T\}\_pi(f)\; xi\; mid\; eta\; angle\; =\; int\_X\; f(x)\; d\; operatorname\{S\}\_pi\; (xi,eta)(x)$

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 {"H"_"x"}_{"x" ∈ "X"} be a μ-measurable family of separable Hilbert spaces. For every "A" ∈ "M", let π("A") be the operator of multiplication by 1_"A" 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 if and only if there is a unitary operator "U":"H" → "K" such that

:$pi(A)\; =\; U^*\; ho(A)\; U\; quad$

for every "A" ∈ "M".

Theorem. If ("X", "M") is a standard Borel space, then for every projection-valued measure π on ("X", "M") taking values in the projections of a "separable" Hilbert space, 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

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

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 taking values in the projections of a separable Hilbert space is an orthogonal direct sum of homogeneous projection-valued measures:

:

where

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

and

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

Generalizations

The idea of a projection-valued measure is generalized by the positive operator-valued measure, where the need for the orthogonality implied by projection operators is replaced by the idea of a set of operators that are a non-orthogonal partition of unity.

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.