PlanetMath: direct integral of Hilbert spaces

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (192)
Orphanage
Unclass'd
Unproven (496)
Corrections (23)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

direct integral of Hilbert spaces

(Definition)

Let be a measure space with measure $\mu$ . To each point $x \in X$ , assign a Hilbert space . Then the direct integral of this family of Hilbert spaces indexed by the points of a measure space, denoted

$\displaystyle \int_{\oplus X} H(x) \, d\mu(x)$

is the set of all functions $v \colon X \to \bigcup_{x \in X} H(x)$ (as usual in functional analysis, we regard two functions that disagree on a set of measure zero as the same) such that

$v(x) \in H(x)$
$\Vert v(x) \Vert \in L^2 (X,\mu)$

Vector addition and scalar multiplication are defined pointwise:

and

. The inner product is defined as

$\displaystyle \langle u, v \rangle = \int_X \langle u(x), v(x) \rangle \, dx \qquad.$

A nice illustration of the direct integral is the expression of $L^2 (\mathbb{R}^2)$ as a direct integral of a continuous infinity of copies of $L^2 (\mathbb{R})$ . Let the space consists of all functions of the form $(x,y) \in \mathbb{R}^2 \mapsto e^{ikx} g(y)$ such that $\int_{-\infty}^{+\infty} \vert g(y)\vert^2 < \infty$ . Define the inner product of two functions $e^{ikx} g_1(y) \in H(k)$ and $e^{ikx} g_2(y) \in H(k)$ as $\int_{-\infty}^{+\infty} g_1 (y) g_2 (y)$ . (By the way, note that is not a subset of $L^2 (\mathbb{R}^2)$ because the functions contained in are not square-integrable in the variable.)

By the definition, the space $\int_{\oplus -\infty}^{+\infty} H(k) \, dk$ consists of functions $(k,x,y) \mapsto e^{ikx} g(k,y)$ such that

$\displaystyle \int_{-\infty}^{+\infty} \Vert g(k)\Vert _{H(k)} = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \vert g(k,y)\vert^2 \,dy \,dk < \infty$

Elements of this space may be regarded as functions of

and

in a natural way:

$\displaystyle \tilde g(x,y) = \int_{-\infty}^{+\infty} e^{ikx} g(k,y)$

By Parseval's identity, $\tilde g \in L^2 (\mathbb{R}^2)$ .

The reason for choosing this example is that it illustrates two situations in which direct integrals are usually encountered. First, direct integrals can be used to provide an analogue of the decomposition of a vector space as a direct sum of eigenspaces of a matrix for operators with continuous spectra. The space in the example can be regarded as the “eigenspace” of the formally self-adjoint operator $-i \partial / \partial x$ with eigenvalue . (It is not eigenspaces in the strict sense of the term since its elements are not square integrable in .) The spectrum of this operator is the whole real axis and the direct integral of these “eigenspaces” is the whole vector space on which the operator acts.

A second situation which can be illustrated with this example is the use of direct integrals in group representation theory. Consider the group of translations along the axis. Each of the spaces transforms under a different representation of this group -- under translation by an amount along the axis, an element of is multiplied by a factor $e^{i k x_0}$ . The direct integral is the infinite-dimensional analogue of the decomposition of a finite-dimensional vector space on which a group acts as a direct sum of irreducible representations.

"direct integral of Hilbert spaces" is owned by rspuzio.

(view preamble)

Also defines:

direct integral

Log in to rate this entry.
(view current ratings)

Cross-references: irreducible, finite-dimensional, infinite-dimensional, factor, representation, Transforms, axis, translations, group, theory, group representation, real axis, spectrum, square, term, strict, eigenvalue, self-adjoint operator, operators, matrix, eigenspaces, direct sum, vector space, decomposition, Parseval's identity, variable, contained, subset, infinity, continuous, expression, inner product, pointwise, multiplication, scalar, vector addition, measure zero, functional analysis, functions, indexed by, Hilbert space, point, measure, measure space
There are 2 references to this entry.

This is version 10 of direct integral of Hilbert spaces, born on 2004-10-12, modified 2004-10-14.
Object id is 6364, canonical name is DirectIntegralOfHilbertSpaces.
Accessed 2883 times total.

Classification:

AMS MSC:

46C05 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Hilbert and pre-Hilbert spaces: geometry and topology )

Pending Errata and Addenda

None.[ View all 1 ]

Discussion

forum policy

No messages.

Interact