Projection-valued measure

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 Tπ("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:

: pi = igoplus_{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}.

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.


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