Spectral Measures and the Krein–Naimark Dilation

Introduction

A spectral measure is a projection-valued measure: it assigns to each measurable set a projection, it is countably additive in the strong operator topology, and the projections multiply by intersection. It is the object into which the spectral theorem of a normal operator condenses, and it is the reason a normal operator is a multiplication by the independent variable. Positive operator-valued measures are its relaxation: the values are positive contractions rather than projections, so the multiplicativity is lost and the additivity is kept. Naimark's dilation theorem asserts that the relaxation is not a genuinely larger class: every positive operator-valued measure on a Hilbert space is the compression of a projection-valued measure on a larger Hilbert space, and the dilating space is minimal and unique up to a unitary. The theorem is the measure-theoretic form of the statement that every operator measure is the shadow of a spectral measure, and it is the reason an operator-valued measure, a priori only positive, can be represented by a projection-valued measure on an enlarged Hilbert space. The adjective Krein–Naimark attaches the theorem to the operator-valued measure theory of Krein and the dilation of Naimark, and to the closely related theorem on the self-adjoint extensions of a symmetric operator via its operator-valued measure.

This article fixes the projection-valued measures and their calculus, the positive operator-valued measures, the Naimark dilation theorem with the minimality and uniqueness, the link with the spectral theorem and the Stinespring dilation, and the Krein–Naimark extension theorem. The spectral theorem for a self-adjoint or normal operator is Self-Adjoint Operators and the Spectral Theorem; the spectral measure on the circle is Unitary Operators and the Spectral Measure; the measure-theoretic integration is Measure Theory and Integration; the indefinite-metric spectral theory is The General Spectral Theorem on a Krein Space; the operator-valued measure theory over a Hilbert algebra is Spectral Measures and the Krein–Naimark Dilation on a Hilbert Algebra (Part II), of which the present article is the bounded-operator reading.

Throughout, $(\Omega,\Sigma)$ is a measurable space, $H$ is a complex Hilbert space with inner product linear in the first argument, $B(H)$ is the algebra of bounded operators, and $E,F$ denote operator-valued measures on $\Sigma$ with values in $B(H)$.

Projection-Valued Measures

Definition. A projection-valued measure (or spectral measure) on $H$ is a map $E:\Sigma\to B(H)$ with

$$ E(\varnothing)=0,\quad E(\Omega)=I,\quad E(\Delta_1\cap\Delta_2)=E(\Delta_1)E(\Delta_2), $$

each $E(\Delta)$ an orthogonal projection, and $E\bigl(\bigcup_n\Delta_n\bigr)=\sum_nE(\Delta_n)$ for pairwise disjoint $\Delta_n$, the sum converging in the strong operator topology.

Proposition (the elementary calculus). For a spectral measure $E$ the following hold: $E(\Delta_1)E(\Delta_2)=0$ for disjoint $\Delta_1,\Delta_2$; $E(\Delta_1\cup\Delta_2)=E(\Delta_1)+E(\Delta_2)$ for disjoint sets; $\Delta_1\subseteq\Delta_2$ implies $E(\Delta_1)\le E(\Delta_2)$; $E(\Delta)^*=E(\Delta)$, and the ranges of the projections together with the strong limits generate a commutative von Neumann algebra.

Proof. These are the algebraic consequences of multiplicativity and additivity: the projection axiom gives the orthogonality, the additivity gives the sum formula and the order, the self-adjointness is the projection property, and the commutativity is the multiplicativity $E(\Delta_1)E(\Delta_2)=E(\Delta_1\cap\Delta_2)=E(\Delta_2)E(\Delta_1)$.

Theorem (integration). For every bounded measurable function $f$ on $\Omega$ the strong integral

$$ \int_\Omega f\,dE=\lim_{\text{refinements}}\sum_kf(\omega_k)E(\Delta_k) $$

exists and defines a bounded operator with

$$ \Bigl\|\int_\Omega f\,dE\Bigr\|\le\sup_{\omega\in\Omega}|f(\omega)| , $$

the map $f\mapsto\int f\,dE$ is an isometric $*$-homomorphism of the algebra of bounded measurable functions onto the von Neumann algebra generated by the $E(\Delta)$, and the measure is recovered from the integral by $E(\Delta)=\int\mathbf{1}_\Delta dE$.

Proof. The integral is defined by approximation with simple functions, the bound and the multiplicativity are inherited from the pointwise operations on simple functions and passed to the limit; the surjectivity onto the generated von Neumann algebra is the spectral theorem for the commutative algebra generated by the projections, and the recovery formula is the specialisation $f=\mathbf{1}_\Delta$.

Proposition (the spectral theorem as a special case). A bounded self-adjoint operator is $T=\int\lambda\,dE(\lambda)$ for a spectral measure on $\sigma(T)\subseteq\mathbb{R}$, and a bounded normal operator is $T=\int z\,dE(z)$ for a spectral measure on $\sigma(T)\subseteq\mathbb{C}$; conversely, for every self-adjoint $A$ in the von Neumann algebra generated by $E$ the measure is the spectral measure of $A$.

Proof. This is the spectral theorem of Self-Adjoint Operators and the Spectral Theorem, read as the construction of the measure from the operator; the converse is the functional calculus applied to $A=f(T)$.

Positive Operator-Valued Measures

Definition. A positive operator-valued measure (or semispectral measure) on $H$ is a map $E:\Sigma\to B(H)$ with

$$ 0\le E(\Delta)\le I\ \ \text{for all}\ \Delta,\qquad E(\Omega)=I,\qquad E\Bigl(\bigcup_n\Delta_n\Bigr)=\sum_nE(\Delta_n) $$

for pairwise disjoint $\Delta_n$, the sum converging in the strong operator topology.

Proposition (the scalar measures). For every $x\in H$ the assignment $\Delta\mapsto\langle E(\Delta)x,x\rangle$ is a finite positive scalar measure with total mass $\|x\|^2$, and $\Delta_1\cap\Delta_2=\varnothing$ implies the positivity of the cross term, so the polarization

$$ \langle E(\Delta)x,y\rangle $$

defines for each pair a complex measure; but the multiplicativity $E(\Delta_1\cap\Delta_2)=E(\Delta_1)E(\Delta_2)$ need not hold.

Proof. The scalar assignment is countably additive and positive because the operator values are between $0$ and $I$; the polarisation is the standard construction of a complex measure from its diagonal; the failure of multiplicativity is the definition of the relaxed class, and the spectral measures are exactly the positive operator-valued measures whose values are projections.

Example (an operator-valued measure that is not a spectral measure). On $H=\mathbb{C}^2$ let $\mu$ be a finite positive measure on $\Omega$ and let $S:\Omega\to B(H)$ be a weakly measurable family of positive contractions; then $E(\Delta)=\int_\Delta S(\omega)\,d\mu(\omega)$ is a positive operator-valued measure. Taking $S(\omega)=\frac12I$ for every $\omega$ gives $E(\Delta)=\frac12\mu(\Delta)I$, whose values are not projections, so $E$ is not a spectral measure; the Naimark dilation below represents it by a projection-valued measure on a larger Hilbert space. The scalar weights $\langle E(\Delta)x,x\rangle$ are positive measures of total mass $\|x\|^2$, and the positivity of $E$ is exactly the positivity of these weights.

Proposition (the Radon–Nikodym derivative). If $E$ is absolutely continuous with respect to a scalar measure $\mu$ then there is a weakly measurable family of positive contractions $S(\omega)$ with $E(\Delta)=\int_\Delta S(\omega)\,d\mu(\omega)$; the family is a spectral density, and $E$ is a spectral measure exactly when $S(\omega)$ is a projection for $\mu$-almost every $\omega$.

Proof. The Radon–Nikodym theorem applied to the scalar measures $\langle E(\cdot)x,y\rangle$ gives the density for each pair, and the densities assemble into the operator family by the polarisation and the boundedness of the measures; the projection criterion is the multiplicativity of $E$ expressed on the densities.

The Naimark Dilation Theorem

Theorem (Naimark). Let $E$ be a positive operator-valued measure on $H$ over the measurable space $(\Omega,\Sigma)$. Then there is a Hilbert space $K$, a spectral measure $F$ on $K$ over $(\Omega,\Sigma)$, and an isometry $V:H\to K$ with

$$ E(\Delta)=V^*F(\Delta)V\qquad(\Delta\in\Sigma). $$

If the dilation is required to be minimal, meaning $K$ is the closed linear span of the subspaces $F(\Delta)VH$ for $\Delta\in\Sigma$, then it is unique up to a unitary of $K$ commuting with the dilation; the minimal dilation is obtained as the closure of the span of the vectors $x\otimes\mathbf{1}_\Delta$ for a suitable direct integral, and the compressed measure is recovered by the formula.

Proof. On the algebraic tensor product $H\otimes L^2(\Omega,\Sigma,\mu)$ with $\mu$ a dominating scalar measure for the family $\langle E(\cdot)x,x\rangle$ define a semi-inner product by $\langle x\otimes\mathbf{1}_\Delta,y\otimes\mathbf{1}_{\Delta'}\rangle=\langle E(\Delta\cap\Delta')x,y\rangle$; the positivity of $E$ makes it positive semidefinite, its null space is a subspace, and the completion $K$ of the quotient carries the multiplication by the characteristic functions as a spectral measure $F$; the map $Vx=x\otimes\mathbf{1}_\Omega$ is an isometry, and $V^*F(\Delta)Vx=E(\Delta)x$ is a direct computation. Minimality and uniqueness are the standard argument for cyclic-vector dilations: two minimal dilations are intertwined by a unitary fixing the image of $V$, which is dense in the minimal span.

Corollary (reduction to the scalar case). The dilation is determined by the scalar measure $\mu$ and the weakly measurable family of positive contractions $S(\omega)$: the dilating space is the closure of the range of the map $x\otimes f\mapsto f(\omega)S(\omega)^{1/2}x$, and the isometry $V$ identifies $H$ with the constant sections, so the dilation is the multiplication by the variable on a space of $H$-valued square-integrable functions.

Proof. Substituting the density $E(\Delta)=\int_\Delta S\,d\mu$ into the semi-inner product shows that the completion is the $L^2$ space of $H$-valued functions with the inner product $\int\langle S(\omega)x,y\rangle\,d\mu$, and the multiplication operator is the spectral measure; the constant section $\mathbf{1}_\Omega\otimes x$ is the image of $x$ under $V$.

Dilation and the Spectral Theorem

Proposition (the spectral measure is its own dilation). A projection-valued measure $E$ on $H$ is its own Naimark dilation, with $K=H$, $F=E$ and $V=I$; conversely, a positive operator-valued measure with a dilation of the same dimension is a spectral measure, so the minimal dilation is trivial exactly for the projection-valued measures.

Proof. The dilation formula with $V=I$ and $F=E$ is the identity of the measure; conversely, if $E(\Delta)=V^*E(\Delta)V$ with $V$ unitary, then the multiplicativity of $E$ is inherited from the multiplicativity of $F$.

Theorem (Stinespring). The Naimark dilation is the commutative case of the Stinespring dilation: a completely positive map $\varphi:\mathcal{A}\to B(H)$ on a $C^*$-algebra has the form $\varphi(a)=V^*\pi(a)V$ for a representation $\pi$ of $\mathcal{A}$ on a Hilbert space $K$ and a bounded linear $V:H\to K$, and the dilation is minimal and unique up to unitary equivalence; when $\mathcal{A}$ is commutative and $\pi$ is the representation of the functions by multiplication, the Stinespring dilation specialises to the Naimark dilation.

Proof. This is the Stinespring theorem; the commutative case is the spectral theorem for the representation $\pi$, whose spectral measure is the $F$ of the Naimark dilation, and the isometry $V$ is the Stinespring operator.

Proposition (the generator of a contraction semigroup). A strongly continuous contraction semigroup on $H$ has a unitary dilation on a larger Hilbert space, and the associated positive operator-valued measure of its generator is dilated by the spectral measure of the unitary group; in this way the dilation theorems undo the compression in which a semigroup is the projection of a group.

Proof. The semigroup satisfies the Hille–Yosida conditions, so its generator $A$ is dissipative and $-A$ is positive; the spectral measure of the self-adjoint part dialates the measure of the semigroup, and the unitary group of the dilation has the semigroup as its compression.

The Krein–Naimark Extension Theorem

Definition. A operator-valued measure for a symmetric operator $T$ is a positive operator-valued measure $\Phi$ on the real line with

$$ \langle TUx,Uy\rangle=\int\lambda\,d\langle\Phi(\lambda)Ux,Uy\rangle $$

for $x,y$ in the domain of $T$ and $U$ an isometry of $H$ into a larger space; the measure is the Krein–Naimark measure of $T$.

Theorem (Krein–Naimark). A densely defined symmetric operator $T$ with equal deficiency indices has a self-adjoint extension, and the extension is described by an operator-valued measure of the above type; conversely, an operator-valued measure satisfying the identity gives a self-adjoint extension by Naimark's dilation, so the self-adjoint extensions of $T$ correspond to the dilations of its operator-valued measures.

Proof. The deficiency indices of a symmetric operator are the dimensions of the deficiency subspaces, and by von Neumann's criterion the self-adjoint extensions correspond to the unitary maps between them, exactly as in The Adjoint of an Unbounded Operator; the operator-valued measure is the compression of the spectral measure of the extension, and Naimark's theorem reverses the construction, so the two classifications agree.

Remark (the definite metric and the indefinite one). In a Krein space the analogue of the spectral measure is a projection-valued measure that is self-adjoint with respect to the indefinite form, and the analogue of the Naimark dilation is the dilation of a positive operator-valued measure with respect to the indefinite form into a spectral measure; the finitely many exceptional points of The General Spectral Theorem on a Krein Space appear as the atoms at which the dilation is not definite.

Summary

A spectral measure, or projection-valued measure, assigns orthogonal projections to the measurable sets with $E(\Delta_1\cap\Delta_2)=E(\Delta_1)E(\Delta_2)$ and countable additivity; its integral is an isometric $*$-homomorphism whose values form a commutative von Neumann algebra, and the spectral theorem of a self-adjoint or normal operator is the special case of integration against the spectral measure on the spectrum. A positive operator-valued measure keeps the positivity $0\le E(\Delta)\le I$ and the countable additivity and drops the multiplicativity; its scalar diagonals are positive measures, so a general operator-valued measure with the scalar weights $\langle E(\Delta)x,x\rangle$ is the relaxation of a spectral measure. Naimark's dilation theorem states that every positive operator-valued measure is the compression $E(\Delta)=V^*F(\Delta)V$ of a spectral measure on a larger Hilbert space, and that the minimal dilation is unique up to unitary equivalence; the dilation is the multiplication by the variable on a space of $H$-valued $L^2$ functions, which is the commutative case of the Stinespring dilation of a completely positive map. In the theory of unbounded operators the same construction is the Krein–Naimark extension theorem, in which the self-adjoint extensions of a symmetric operator correspond to the dilations of its operator-valued measures, and the indefinite-metric version has its atoms at the finitely many exceptional points of the Krein-space spectral theorem.

Summary of Notation

Symbol	Meaning
$E(\Delta)$, $F(\Delta)$	operator-valued measures on $(\Omega,\Sigma)$
$E(\Delta_1\cap\Delta_2)=E(\Delta_1)E(\Delta_2)$	multiplicativity, spectral measure
$0\le E(\Delta)\le I$	positive operator-valued measure
$\int f\,dE$	integration, $*$-homomorphism
$E(\Delta)=V^*F(\Delta)V$	Naimark dilation
$K\supseteq H$, $V$	dilating space and isometry
$V^*\pi(a)V$	Stinespring dilation
minimality of the dilation	uniqueness up to unitary
$\Phi$ for a symmetric operator	Krein–Naimark measure
extension $\leftrightarrow$ dilation	Krein–Naimark theorem