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.

1 Formal definition
2 Extensions of projection-valued measures
3 Structure of projection-valued measures
4 Generalizations
5 References

[edit] Formal definition

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

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

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

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

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

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 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 T_π(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 {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

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

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 {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:

$\pi = \bigoplus_{1 \leq n \leq \omega} (\pi | H_n)$

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 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.

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Categories: Linear algebra | Spectral theory | Measures (measure theory)