Projection-valued measure

Projection-Valued Measure

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

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

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

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

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

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

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

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

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

<math> mathbf_A mapsto......

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