Projection-Valued Measure

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In mathematics, particularly functional analysis a **projection-valued measure (PVM)** 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

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

is a complex measure on*M* (that is, a complex-valued countably additive function). We denote this measure by <math>operatorname_pi(xi, eta)</math>.

If π is a projection-valued measure and

then π(*A*), π(*B*) are orthogonal projections. From this follows that in general,

**Example**. Suppose (*X*, *M*, μ) is a measure space. Let π(*A*) be the operator of multiplication by the indicator function 1<sub>*A*</sub> on *L*<sup>2</sup>. 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

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- <math>pi(X) = operatorname_H quad </math>

and for every ξ, η ∈

- <math>A mapsto langle pi(A)xi mid eta rangle </math>

is a complex measure on

If π is a projection-valued measure and

- <math>A cap B = emptyset,</math>

then π(

- <math> pi(A) pi(B) = pi(A cap B). </math>

- <math> mathbf_A mapsto...... ...

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