Linear algebra | Measures (measure theory) | Spectral theory

Projection-valued measure

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. (Wikipedia).

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Related pages

If and only if | Functional analysis | Self-adjoint operator | Spectral theorem | Indicator function | Step function | POVM | Density matrix | Borel functional calculus | Spectral theory of normal C*-algebras | Measurable space | Mathematics | Complex measure | Function (mathematics) | Ring homomorphism | Self-adjoint | Measurable function | Hilbert space | Σ-algebra | Projection (mathematics) | Lp space | Measure (mathematics) | Spectral theory of compact operators | Direct integral | Spectral theory