In mathematics, particularly in functional analysis, a projection-valued measure (or spectral measure) is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space.[1] A projection-valued measure (PVM) is formally similar to a real-valued measure, except that its 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, in which case the PVM is sometimes referred to as the spectral measure. 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.[clarification needed] 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.


Let denote a separable complex Hilbert space and a measurable space consisting of a set and a Borel σ-algebra on . A projection-valued measure is a map from to the set of bounded self-adjoint operators on satisfying the following properties:[2][3]

The second and fourth property show that if and are disjoint, i.e., , the images and are orthogonal to each other.

Let and its orthogonal complement denote the image and kernel, respectively, of . If is a closed subspace of then can be wrtitten as the orthogonal decomposition and is the unique identity operator on satisfying all four properties.[4][5]

For every and the projection-valued measure forms a complex-valued measure on defined as

with total variation at most .[6] It reduces to a real-valued measure when

and a probability measure when is a unit vector.

Example Let be a σ-finite measure space and, for all , let

be defined as

i.e., as multiplication by the indicator function on L2(X). Then defines a projection-valued measure.[6] For example, if , , and there is then the associated complex measure which takes a measurable function and gives the integral

Extensions of projection-valued measures

If π 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. In fact, it is easy to check that this map is a ring homomorphism. This map extends in a canonical way to all bounded complex-valued measurable functions on X, and we have the following.

Theorem — For any bounded Borel function on , there exists a unique bounded operator such that [7][8]

where is a finite Borel measure given by

Hence, is a finite measure space.

The theorem is also correct for unbounded measurable functions but then will be an unbounded linear operator on the Hilbert space .

This allows to define the Borel functional calculus for such operators and then pass to measurable functions via the Riesz–Markov–Kakutani representation theorem. That is, if is a measurable function, then a unique measure exists such that

Spectral theorem

See also: Self-adjoint operator § Spectral theorem

Let be a separable complex Hilbert space, be a bounded self-adjoint operator and the spectrum of . Then the spectral theorem says that there exists a unique projection-valued measure , defined on a Borel subset , such that[9]

where the integral extends to an unbounded function when the spectrum of is unbounded.[10]

Direct integrals

First we provide a general example of projection-valued measure based on direct integrals. Suppose (X, M, μ) is a measure space and let {Hx}xX be a μ-measurable family of separable Hilbert spaces. For every EM, let π(E) be the operator of multiplication by 1E on the Hilbert space

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

Suppose π, ρ are projection-valued measures on (X, M) with values in the projections of H, K. π, ρ are unitarily equivalent if and only if there is a unitary operator U:HK such that

for every EM.

Theorem. If (X, M) is a standard Borel space, then for every projection-valued measure π on (X, M) taking values in the projections of a separable Hilbert space, there is a Borel measure μ and a μ-measurable family of Hilbert spaces {Hx}xX , such that π is unitarily equivalent to multiplication by 1E on the Hilbert space

The measure class[clarification needed] of μ and the measure equivalence class of the multiplicity function x → dim Hx completely characterize the projection-valued measure up to unitary equivalence.

A projection-valued measure π is homogeneous of multiplicity n if and only if the multiplicity function has constant value n. Clearly,

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



Application in quantum mechanics

See also: Expectation value (quantum mechanics)

In quantum mechanics, given a projection valued measure of a measurable space X to the space of continuous endomorphisms upon a Hilbert space H,

A common choice for X is the real line, but it may also be

Let E be a measurable subset of the measurable space X and Φ a normalized vector-state in H, so that its Hilbert norm is unitary, ||Φ|| = 1. The probability that the observable takes its value in the subset E, given the system in state Φ, is

where the latter notation is preferred in physics.

We can parse this in two ways.

First, for each fixed E, the projection π(E) is a self-adjoint operator on H whose 1-eigenspace is the states Φ for which the value of the observable always lies in E, and whose 0-eigenspace is the states Φ for which the value of the observable never lies in E.

Second, for each fixed normalized vector state , the association

is a probability measure on X making the values of the observable into a random variable.

A measurement that can be performed by a projection-valued measure π is called a projective measurement.

If X is the real number line, there exists, associated to π, a Hermitian operator A defined on H by

which takes the more readable form

if the support of π is a discrete subset of R.

The above operator A is called the observable associated with the spectral measure.


The idea of a projection-valued measure is generalized by the positive 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[clarification needed]. This generalization is motivated by applications to quantum information theory.

See also


  1. ^ Conway 2000, p. 41.
  2. ^ Hall 2013, p. 138.
  3. ^ Reed & Simon 1980, p. 234.
  4. ^ Rudin 1991, p. 308.
  5. ^ Hall 2013, p. 541.
  6. ^ a b Conway 2000, p. 42.
  7. ^ Kowalski, Emmanuel (2009), Spectral theory in Hilbert spaces (PDF), ETH Zürich lecture notes, p. 50
  8. ^ Reed & Simon 1980, p. 227,235.
  9. ^ Reed & Simon 1980, p. 235.
  10. ^ Hall 2013, p. 205.