projection-valued measure, or spectral measure, is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections...

operator-valued measure (POVM) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalization of projection-valued...

by the associated projection operator, and the collection of all the subspaces is then represented by a projection-valued measure. One formulation of...

Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in...

defined, either as a projection-valued measure Ω T {\displaystyle \Omega _{T}} , or as a one-parameter family of projection-valued measures { Σ λ } {\displaystyle...

adjoint, E is said to be a projection-valued measure or PVM.) The theorem reads as follows: Let E be a positive L(H)-valued measure on X. There exists a Hilbert...

called the projection postulate. A more general formulation replaces the projection-valued measure with a positive-operator valued measure (POVM). To...

theorem proves the existence of a certain projection-valued measure (PVM) Q {\displaystyle Q} , the spectral measure of A {\displaystyle A} . In this case:...

formulation of direct sum decomposition is formulated in terms of projection-valued measures. Mackey's original formulation was expressed in terms of a locally...

a Hilbert space H. A has a spectral decomposition, which is a projection-valued measure E defined on the Borel subsets of R. In particular, for any bounded...

discrete, the projection valued measure reduces to a family {Eλ} indexed on some parameter λ ranging over a countable set. Similarly, a Y valued observable...

measurements. That theory was based in turn on the theory of projection-valued measures for self-adjoint operators that had been recently developed (by...

{\displaystyle A=\int a\,dP(a)} with a projection-valued measure P {\displaystyle P} . For the expectation value of A {\displaystyle A} in a pure state...

of Borel measure Structure theorem for Gaussian measures – Mathematical theorem Projection-valued measure – Mathematical operator-value measure of interest...

Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in...

the normal operator N gives rise to a projection-valued measure P on its spectrum, σ(N), which assigns a projection PΩ to each Borel subset of σ(N). N can...

This is a summary of map projections that have articles of their own on Wikipedia or that are otherwise notable. Because there is no limit to the number...

or equal-area projection is a map projection that preserves relative area measure between any and all map regions. Equivalent projections are widely used...

countable chains. A POVM is a projection-valued measure precisely when its image is contained in the orthoalgebra of projections on the Hilbert space H {\displaystyle...

terms of self-adjoint projections on a separable complex Hilbert space H, that is, in terms of a PVM (Projection-valued measure). In the general case...

gross value added (GVA) is the measure of the value of goods and services produced in an area, industry or sector of an economy. "The gross value added...

In linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism)...

a point is the signed measure of its projection on the primary axis. Its absolute value is the distance between the projection and the origin of the axis...

legend from Proctor Valley, east of San Diego, California Projection-valued measure, a type of measure used in functional analysis Perfetti Van Melle, an Italian-Dutch...

Riesz-Markov Theorem, τ {\displaystyle \tau } gives rise to a projection-valued measure on R {\displaystyle \mathbb {R} } that is the resolution of the...

The Mercator projection (/mərˈkeɪtər/) is a conformal cylindrical map projection first presented by Flemish geographer and mapmaker Gerardus Mercator...

A 3D projection (or graphical projection) is a design technique used to display a three-dimensional (3D) object on a two-dimensional (2D) surface. These...

is widely applicable to tomography, the creation of an image from the projection data associated with cross-sectional scans of an object. If a function...

linear dependent, a positive operator-valued measure (POVM) is a natural generalization of a projection-valued measure (PVM) in that elements of a POVM are...

Throughout, H {\displaystyle H} is a fixed Hilbert space. A projection-valued measure on a measurable space ( X , Ω ) , {\displaystyle (X,\Omega ),}...