- Projection-valued measure
In
mathematics , particularlyfunctional analysis a projection-valued measure is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on aHilbert space . Projection-valued measures are used to express results inspectral theory , such as the spectral theorem forself-adjoint operator s.Formal definition
A projection-valued measure on a
measurable space ("X", "M"), where "M" is aσ-algebra of subsets of "X", is a mapping π from "M" to the set ofself-adjoint projections on aHilbert 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 projection s. 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 integral s. 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-orthogonalpartition 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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