Inmathematics, particularly infunctional analysis, aprojection-valued measure, orspectral measure, is a function defined on certain subsets of a fixed set and whose values areself-adjointprojections on a fixedHilbert space.^[1] A projection-valued measure (PVM) is formally similar to areal-valuedmeasure, except that its values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible tointegratecomplex-valued functions with respect to a PVM; the result of such an integration is alinear operator on the given Hilbert space.

Projection-valued measures are used to express results inspectral theory, such as the importantspectral theorem forself-adjoint operators, in which case the PVM is sometimes referred to as thespectral measure. TheBorel functional calculus for self-adjoint operators is constructed using integrals with respect to PVMs. Inquantum mechanics, PVMs are the mathematical description ofprojective measurements.^{[clarification needed]} They are generalized bypositive operator valued measures (POVMs) in the same sense that amixed state ordensity matrix generalizes the notion of apure state.

Let${\displaystyle H}$  denote aseparablecomplexHilbert space and${\displaystyle (X,M)}$  ameasurable space consisting of a set${\displaystyle X}$  and aBorel σ-algebra${\displaystyle M}$  on${\displaystyle X}$ . Aprojection-valued measure${\displaystyle \pi }$  is a map from${\displaystyle M}$  to the set ofbounded self-adjoint operators on${\displaystyle H}$  satisfying the following properties:^[2][3]

• ${\displaystyle \pi (E)}$  is anorthogonal projection for all${\displaystyle E\in M.}$ 
• ${\displaystyle \pi (\mathrm{\varnothing })=0}$  and${\displaystyle \pi (X)=I}$ , where${\displaystyle \mathrm{\varnothing }}$  is theempty set and${\displaystyle I}$  theidentity operator.
• If${\displaystyle E_1,E_2,E_3,\dots }$  in${\displaystyle M}$  are disjoint, then for all${\displaystyle v\in H}$ ,

${\displaystyle \pi \left(\bigcup _{j=1}^{\mathrm{\infty }}{E}_j\right)v=\sum _{j=1}^{\mathrm{\infty }}\pi (E_j)v.}$ 

• ${\displaystyle \pi (E_1\cap E_2)=\pi (E_1)\pi (E_2)}$  for all${\displaystyle E_1,E_2\in M.}$ 

The second and fourth property show that if${\displaystyle E_1}$  and${\displaystyle E_2}$  are disjoint, i.e.,${\displaystyle E_1\cap E_2=\mathrm{\varnothing }}$ , the images${\displaystyle \pi (E_1)}$  and${\displaystyle \pi (E_2)}$  areorthogonal to each other.

Let${\displaystyle V_E=\mathrm{im}(\pi (E))}$  and itsorthogonal complement${\displaystyle V_E^{\perp }=\ker (\pi (E))}$  denote theimage andkernel, respectively, of${\displaystyle \pi (E)}$ . If${\displaystyle V_E}$  is a closed subspace of${\displaystyle H}$  then${\displaystyle H}$  can be wrtitten as theorthogonal decomposition${\displaystyle H=V_E\oplus V_E^{\perp }}$  and${\displaystyle \pi (E)=I_E}$  is the unique identity operator on${\displaystyle V_E}$  satisfying all four properties.^[4][5]

For every${\displaystyle \xi ,\eta \in H}$  and${\displaystyle E\in M}$  the projection-valued measure forms acomplex-valued measure on${\displaystyle H}$  defined as

${\displaystyle {\mu }_{\xi ,\eta }(E):=⟨\pi (E)\xi \mid \eta ⟩}$ 

withtotal variation at most${\displaystyle \Vert \xi \Vert \Vert \eta \Vert }$ .^[6] It reduces to a real-valuedmeasure when

${\displaystyle {\mu }_{\xi }(E):=⟨\pi (E)\xi \mid \xi ⟩}$ 

and aprobability measure when${\displaystyle \xi }$  is aunit vector.

Example Let${\displaystyle (X,M,\mu )}$  be aσ-finite measure space and, for all${\displaystyle E\in M}$ , let

${\displaystyle \pi (E):L^2(X)\to L^2(X)}$ 

be defined as

${\displaystyle \psi \mapsto \pi (E)\psi =1_E\psi ,}$ 

i.e., as multiplication by theindicator function${\displaystyle 1_E}$  onL²(X). Then${\displaystyle \pi (E)=1_E}$  defines a projection-valued measure.^[6] For example, if${\displaystyle X=\mathbb{R}}$ ,${\displaystyle E=(0,1)}$ , and${\displaystyle \phi ,\psi \in L^2(\mathbb{R})}$  there is then the associated complex measure${\displaystyle {\mu }_{\phi ,\psi }}$  which takes a measurable function${\displaystyle f:\mathbb{R}\to \mathbb{R}}$  and gives the integral

${\displaystyle {\int }_{E}fd{\mu }_{\phi ,\psi }={\int }_0^{1}f(x)\psi (x)\overline{\phi }(x)dx}$ 

Ifπ is a projection-valued measure on a measurable space (X,M), then the map

${\displaystyle {\chi }_E\mapsto \pi (E)}$ 

extends to a linear map on the vector space ofstep functions onX. In fact, it is easy to check that this map is aring homomorphism. This map extends in a canonical way to all bounded complex-valuedmeasurable functions onX, and we have the following.

The theorem is also correct for unbounded measurable functions${\displaystyle f}$  but then${\displaystyle T}$  will be an unbounded linear operator on the Hilbert space${\displaystyle H}$ .

This allows to define theBorel functional calculus for such operators and then pass to measurable functions via theRiesz–Markov–Kakutani representation theorem. That is, if${\displaystyle g:\mathbb{R}\to \mathbb{C}}$  is a measurable function, then a unique measure exists such that

${\displaystyle g(T):={\int }_{\mathbb{R}}g(x)d\pi (x).}$ 

Let${\displaystyle H}$  be aseparablecomplexHilbert space,${\displaystyle A:H\to H}$  be a boundedself-adjoint operator and${\displaystyle \sigma (A)}$  thespectrum of${\displaystyle A}$ . Then thespectral theorem says that there exists a unique projection-valued measure${\displaystyle {\pi }^A}$ , defined on aBorel subset${\displaystyle E\subset \sigma (A)}$ , such that^[9]

${\displaystyle A={\int }_{\sigma (A)}\lambda d{\pi }^A(\lambda ),}$ 

where the integral extends to an unbounded function${\displaystyle \lambda }$  when the spectrum of${\displaystyle A}$  is unbounded.^[10]

First we provide a general example of projection-valued measure based ondirect integrals. Suppose (X,M, μ) is a measure space and let {H_x}_{x ∈X} be a μ-measurable family of separable Hilbert spaces. For everyE ∈M, letπ(E) be the operator of multiplication by 1_E on the Hilbert space

${\displaystyle {\int }_X^{\oplus }{H}_{x}d\mu (x).}$ 

Thenπ is a projection-valued measure on (X,M).

Supposeπ, ρ are projection-valued measures on (X,M) with values in the projections ofH,K.π, ρ areunitarily equivalentif and only if there is a unitary operatorU:H →K such that

${\displaystyle \pi (E)=U^{\ast }\rho (E)U}$ 

for everyE ∈M.

Theorem. If (X,M) is astandard Borel space, then for every projection-valued measureπ on (X,M) taking values in the projections of aseparable Hilbert space, there is a Borel measure μ and a μ-measurable family of Hilbert spaces {H_x}_{x ∈X}, such thatπ is unitarily equivalent to multiplication by 1_E on the Hilbert space

${\displaystyle {\int }_X^{\oplus }{H}_{x}d\mu (x).}$ 

The measure class^{[clarification needed]} of μ and the measure equivalence class of the multiplicity functionx → dimH_x completely characterize the projection-valued measure up to unitary equivalence.

A projection-valued measureπ ishomogeneous of multiplicityn if and only if the multiplicity function has constant valuen. Clearly,

Theorem. Any projection-valued measureπ taking values in the projections of a separable Hilbert space is an orthogonal direct sum of homogeneous projection-valued measures:

${\displaystyle \pi =\underset{1\le n\le \omega }{⨁}(\pi \mid H_n)}$ 

where

${\displaystyle H_n={\int }_{X_n}^{\oplus }{H}_{x}d(\mu \mid X_n)(x)}$ 

and

${\displaystyle X_n=\{x\in X:\dim H_x=n\}.}$ 

In quantum mechanics, given a projection-valued measure of a measurable space${\displaystyle X}$  to the space of continuous endomorphisms upon a Hilbert space${\displaystyle H}$ ,

• theprojective space${\displaystyle \mathbf{P}(H)}$  of the Hilbert space${\displaystyle H}$  is interpreted as the set of possible (normalizable) states${\displaystyle \phi }$  of a quantum system,^[11]
• the measurable space${\displaystyle X}$  is the value space for some quantum property of the system (an "observable"),
• the projection-valued measure${\displaystyle \pi }$  expresses the probability that theobservable takes on various values.

A common choice for${\displaystyle X}$  is the real line, but it may also be

• ${\displaystyle {\mathbb{R}}^3}$  (for position or momentum in three dimensions ),
• a discrete set (for angular momentum, energy of a bound state, etc.),
• the 2-point set "true" and "false" for the truth-value of an arbitrary proposition about${\displaystyle \phi }$ .

Let${\displaystyle E}$  be a measurable subset of${\displaystyle X}$  and${\displaystyle \phi }$  a normalizedvector quantum state in${\displaystyle H}$ , so that its Hilbert norm is unitary,${\displaystyle \Vert \phi \Vert =1}$ . The probability that the observable takes its value in${\displaystyle E}$ , given the system in state${\displaystyle \phi }$ , is

${\displaystyle P_{\pi }(\phi )(E)=⟨\phi \mid \pi (E)(\phi )⟩=⟨\phi \mid \pi (E)\mid \phi ⟩.}$ 

We can parse this in two ways. First, for each fixed${\displaystyle E}$ , the projection${\displaystyle \pi (E)}$  is aself-adjoint operator on${\displaystyle H}$  whose 1-eigenspace are the states${\displaystyle \phi }$  for which the value of the observable always lies in${\displaystyle E}$ , and whose 0-eigenspace are the states${\displaystyle \phi }$  for which the value of the observable never lies in${\displaystyle E}$ .

Second, for each fixed normalized vector state${\displaystyle \phi }$ , the association

${\displaystyle P_{\pi }(\phi ):E\mapsto ⟨\phi \mid \pi (E)\phi ⟩}$ 

is a probability measure on${\displaystyle X}$  making the values of the observable into a random variable.

A measurement that can be performed by a projection-valued measure${\displaystyle \pi }$  is called aprojective measurement.

If${\displaystyle X}$  is the real number line, there exists, associated to${\displaystyle \pi }$ , a self-adjoint operator${\displaystyle A}$  defined on${\displaystyle H}$  by

${\displaystyle A(\phi )={\int }_{\mathbb{R}}\lambda d\pi (\lambda )(\phi ),}$ 

which reduces to

${\displaystyle A(\phi )=\sum _i{\lambda }_i\pi ({\lambda }_i)(\phi )}$ 

if the support of${\displaystyle \pi }$  is a discrete subset of${\displaystyle X}$ .

The above operator${\displaystyle A}$  is called the observable associated with the spectral measure.

The idea of a projection-valued measure is generalized by thepositive operator-valued measure (POVM), where the need for the orthogonality implied by projection operators is replaced by the idea of a set of operators that are a non-orthogonal "partition of unity", i.e. a set ofpositive semi-definiteHermitian operators that sum to the identity. This generalization is motivated by applications toquantum information theory.

• Spectral theorem
• Spectral theory of compact operators
• Spectral theory of normal C*-algebras

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