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spectral measure

(Definition)

Definition and Properties

In this entry by a projection we mean an orthogonal projection over some Hilbert space.

Let be an Hilbert space and the algebra of bounded operators in endowed with the strong operator topology.

Let $\mathcal{B}$ denote the Borel $\sigma$ -algebra of $\mathbb{C}$ .

Definition - A spectral measure is a function $P : \mathcal{B} \longrightarrow B(H)$ such that

$P(\emptyset)=0$
$P(\mathbb{C}) = I$ , where denotes the identity operator in .
is a projection in for every $E \in \mathcal{B}$
$\displaystyle P(\bigcup_{n=1}^{\infty} E_n) = \sum_{n=1}^{\infty} P(E_n)\;\;$ for every sequence $E_1, E_2, \ldots$ of disjoint sets in $\mathcal{B}$ , where the sum on the right is interpreted as the limit in the strong operator topology.

The fact that the limit in the last condition exists and is a projection is a consequence of the Banach-Steinhaus theorem and the following properties (that depend only on the previous conditions):

Properties : In the following denotes the range of an operator $T \in B(H)$ .

$E_1 \subseteq E_2\; \Longrightarrow \; Ran(P(E_1)) \subseteq Ran(P(E_2))$ .
$E_1 \cap E_2 = \emptyset \; \Longrightarrow \; Ran(P(E_1)) \perp Ran(P(E_2))$ .
$P(E_1 \cap E_2) = P(E_1)P(E_2)\;\;$ for every $E_1, E_2 \in \mathcal{B}$ .

Thus, a spectral measure is a countably additive vector measure that takes projections as values. For that, spectral measures are also called projection valued measures.

Integration against spectral measures

Let $f :\mathbb{C} \longrightarrow \mathbb{C}$ be a bounded Borel function and a spectral measure. We are interested to give meaning to the integral

$\displaystyle \int_{\mathbb{C}}f dP$

We are now going to outline how that can be done:

First we notice that for every vectors $\xi, \eta \in H$ we can define a complex measure $\mu_{\xi, \eta}$ by

$\displaystyle \mu_{\xi, \eta}(E) = \langle P(E)\xi, \eta \rangle,$

whose total variation is estimated by $\Vert\mu_{\xi, \eta}\Vert \leq \Vert\xi\Vert \Vert\eta\Vert$ .

Then we notice that the function $B : H \times H \longrightarrow \mathbb{C}$ defined by

$\displaystyle B(\xi, \eta) :=\int_{\mathbb{C}} f d\mu_{\xi, \eta}$

is a bounded sesquilinear form.

By the Riesz representation theorem, there exists a unique operator $\pi(f) \in B(H)$ such that

$\displaystyle \langle\pi(f) \xi, \eta \rangle = \int_{\mathbb{C}} f d\mu_{\xi, \eta}, \quad\quad \xi, \eta \in H$

We can then take $\displaystyle \int_{\mathbb{C}} f dP := \pi(f)$ .

The above definition can look rather artificial and does not reveal why should it be a “good” definition of $\displaystyle \int_{\mathbb{C}} f dP$ . That is clarified by the following properties:

Properties :

$\displaystyle \int_{\mathbb{C}} \chi_E \; dP = P(E)$ for every characteristic function $\chi_E$ .
The map $\chi_E \longmapsto P(E)$ extends continuously to a homomorphism defined on all bounded borel functions
$\displaystyle f \longmapsto \int_{\mathbb{C}} f dP$

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Other names:

projection valued measure

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Cross-references: homomorphism, map, characteristic function, properties, Riesz representation theorem, sesquilinear form, total variation, complex measure, vectors, integral, countably additive vector measure, operator, consequence, limit, disjoint, sequence, identity operator, function, strong operator topology, bounded operators, algebra, Hilbert space, orthogonal projection
There are 3 references to this entry.

This is version 5 of spectral measure, born on 2007-09-12, modified 2007-09-15.
Object id is 9932, canonical name is SpectralMeasure.
Accessed 1107 times total.

Classification:

AMS MSC:	28B05 (Measure and integration :: Set functions, measures and integrals with values in abstract spaces :: Vector-valued set functions, measures and integrals)
	28C20 (Measure and integration :: Set functions and measures on spaces with additional structure :: Set functions and measures and integrals in infinite-dimensional spaces )
	46G10 (Functional analysis :: Measures, integration, derivative, holomorphy :: Vector-valued measures and integration)
	46G12 (Functional analysis :: Measures, integration, derivative, holomorphy :: Measures and integration on abstract linear spaces)
	47A56 (Operator theory :: General theory of linear operators :: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)

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