Projection-valued measure

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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.

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[edit] 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 \rangle

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 1A on L2(X). Then π is a projection-valued measure.

[edit] 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 \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.

[edit] 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 separable 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 if and only if 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) taking values in the projections of a separable Hilbert space, 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 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 = \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\}.

[edit] 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.

[edit] 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.