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Concept
Projection-valued measure
Summary
In mathematics, particularly in 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 fixed Hilbert space. Projection-valued measures are formally similar to real-valued measures, except that their values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space.
Projection-valued measures are used to express results in spectral theory, such as the important spectral theorem for self-adjoint operators. The Borel functional calculus for self-adjoint operators is constructed using integrals with respect to PVMs. In quantum mechanics, PVMs are the mathematical description of projective measurements. They are generalized by positive operator valued measures (POVMs) in the same sense that a mixed state or density matrix generalizes the notion of a pure state.
A projection-valued measure on a measurable space
where is a σ-algebra of subsets of , is a mapping from to the set of self-adjoint projections on a Hilbert space (i.e. the orthogonal projections) such that
(where is the identity operator of ) and for every , the following function
is a complex measure on (that is, a complex-valued countably additive function).
We denote this measure by
Note that is a real-valued measure, and a probability measure when has length one.
If is a projection-valued measure and
then the images , are orthogonal to each other. From this follows that in general,
and they commute.
Example. Suppose is a measure space. Let, for every measurable subset in ,
be the operator of multiplication by the indicator function on L2(X). Then is a projection-valued measure. For example, if , , and there is then the associated complex measure which takes a measurable function and gives the integral
If pi is a projection-valued measure on a measurable space (X, M), then the map
extends to a linear map on the vector space of step functions on X.
Official source
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