Let ${\displaystyle H}$  denote a separable complex Hilbert space and ${\displaystyle (X,M)}$  a measurable space consisting of a set ${\displaystyle X}$  and a Borel σ-algebra ${\displaystyle M}$  on ${\displaystyle X}$ . A projection-valued measure ${\displaystyle \pi }$  is a map from ${\displaystyle M}$  to the set of bounded self-adjoint operators on ${\displaystyle H}$  satisfying the following properties:^[2][3]

• ${\displaystyle \pi (E)}$  is an orthogonal projection for all ${\displaystyle E\in M.}$ 
• ${\displaystyle \pi (\emptyset )=0}$  and ${\displaystyle \pi (X)=I}$ , where ${\displaystyle \emptyset }$  is the empty set and ${\displaystyle I}$  the identity operator.
• If ${\displaystyle E_{1},E_{2},E_{3},\dotsc }$  in ${\displaystyle M}$  are disjoint, then for all ${\displaystyle v\in H}$ ,

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

• ${\displaystyle \pi (E_{1}\cap E_{2})=\pi (E_{1})\pi (E_{2})}$  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., ${\displaystyle E_{1}\cap E_{2}=\emptyset }$ , the images ${\displaystyle \pi (E_{1})}$  and ${\displaystyle \pi (E_{2})}$  are orthogonal to each other.

Let ${\displaystyle V_{E}=\operatorname {im} (\pi (E))}$  and its orthogonal complement ${\displaystyle V_{E}^{\perp }=\ker(\pi (E))}$  denote the image and kernel, respectively, of ${\displaystyle \pi (E)}$ . If ${\displaystyle V_{E}}$  is a closed subspace of ${\displaystyle H}$  then ${\displaystyle H}$  can be wrtitten as the orthogonal 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 a complex-valued measure on ${\displaystyle H}$  defined as

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

with total variation at most ${\displaystyle \|\xi \|\|\eta \|}$ .^[6] It reduces to a real-valued measure when

${\displaystyle \mu _{\xi }(E):=\langle \pi (E)\xi \mid \xi \rangle }$ 

and a probability measure when ${\displaystyle \xi }$  is a unit vector.

Example Let ${\displaystyle (X,M,\mu )}$  be a σ-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 the indicator function ${\displaystyle 1_{E}}$  on L²(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

${\displaystyle \int _{E}f\,d\mu _{\phi ,\psi }=\int _{0}^{1}f(x)\psi (x){\overline {\phi }}(x)\,dx}$ 

If π 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 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.

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 the Borel functional calculus for such operators and then pass to measurable functions via the Riesz–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 a separable complex Hilbert space, ${\displaystyle A:H\to H}$  be a bounded self-adjoint operator and ${\displaystyle \sigma (A)}$  the spectrum of ${\displaystyle A}$ . Then the spectral theorem says that there exists a unique projection-valued measure ${\displaystyle \pi ^{A}}$ , defined on a Borel 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 on direct 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 every E ∈ M, let π(E) be the operator of multiplication by 1_E on the Hilbert space

${\displaystyle \int _{X}^{\oplus }H_{x}\ d\mu (x).}$ 

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:H → K such that

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

for every E ∈ M.

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 {H_x}_{x ∈ X} , such that π 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 function x → dim H_x 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:

${\displaystyle \pi =\bigoplus _{1\leq n\leq \omega }(\pi \mid H_{n})}$ 

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 ${\displaystyle H}$ ,

• the projective space ${\displaystyle \mathbf {P} (H)}$  of the Hilbert space ${\displaystyle H}$  is interpreted as the set of possible (normalizable) states ${\displaystyle \varphi }$  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 the observable 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 ${\displaystyle \varphi }$ .

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

${\displaystyle P_{\pi }(\varphi )(E)=\langle \varphi \mid \pi (E)(\varphi )\rangle =\langle \varphi |\pi (E)|\varphi \rangle .}$ 

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

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

${\displaystyle P_{\pi }(\varphi ):E\mapsto \langle \varphi \mid \pi (E)\varphi \rangle }$ 

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 a projective 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

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

which reduces to

${\displaystyle A(\varphi )=\sum _{i}\lambda _{i}\pi ({\lambda _{i}})(\varphi )}$ 

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 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.

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

• Ashtekar, Abhay; Schilling, Troy A. (1999). "Geometrical Formulation of Quantum Mechanics". On Einstein's Path. New York, NY: Springer New York. arXiv:gr-qc/9706069. doi:10.1007/978-1-4612-1422-9_3. ISBN 978-1-4612-7137-6.* Conway, John B. (2000). A course in operator theory. Providence (R.I.): American mathematical society. ISBN 978-0-8218-2065-0.
• Hall, Brian C. (2013). Quantum Theory for Mathematicians. New York: Springer Science & Business Media. ISBN 978-1-4614-7116-5.
• Mackey, G. W., The Theory of Unitary Group Representations, The University of Chicago Press, 1976
• Moretti, Valter (2017), Spectral Theory and Quantum Mechanics Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation, vol. 110, Springer, Bibcode:2017stqm.book.....M, ISBN 978-3-319-70705-1
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
• Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Vol 1: Functional analysis. Academic Press. ISBN 978-0-12-585050-6.
• Rudin, Walter (1991). Functional Analysis. Boston, Mass.: McGraw-Hill Science, Engineering & Mathematics. ISBN 978-0-07-054236-5.
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
• G. Teschl, Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators, https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, American Mathematical Society, 2009.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
• Varadarajan, V. S., Geometry of Quantum Theory V2, Springer Verlag, 1970.