direct integral of Hilbert spaces

direct integral of Hilbert spaces

Let

X

be a measure space with measure

μ

. To each point

x

∈

X

, assign a

Hilbert space

H

⁢

(

x

)

. Then the

direct integral

of this family of Hilbert spaces indexed by the points of a measure space, denoted

∫

⊕

X

H

⁢

(

x

)

⁢

𝑑

μ

⁢

(

x

)

is the set of all functions

v

:

X

→

⋃

x

∈

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

)

∈

H

⁢

(

x

)

•

∥

v

⁢

(

x

)

∥

∈

L

2

⁢

(

X

,

μ

)

Vector addition

and

scalar multiplication

are defined pointwise:

(

u

+

v

)

⁢

(

x

)

=

u

⁢

(

x

)

+

v

⁢

(

x

)

and

(

s

⁢

v

)

⁢

(

x

)

=

s

⁢

v

⁢

(

x

)

. The

inner product

is defined as

⟨

u

,

v

⟩

=

∫

X

⟨

u

⁢

(

x

)

,

v

⁢

(

x

)

⟩

⁢

𝑑

x

.

A nice illustration of the direct integral is the

expression

of

L

2

⁢

(

ℝ

2

)

as a direct integral of a

continuous

infinity of copies of

L

2

⁢

(

ℝ

)

. Let the space

H

⁢

(

k

)

consists of all functions of the form

(

x

,

y

)

∈

ℝ

2

↦

e

i

⁢

k

⁢

x

⁢

g

⁢

(

y

)

such that

∫

-

∞

+

∞

|

g

⁢

(

y

)

|

2

<

∞

. Define the inner product of two functions

e

i

⁢

k

⁢

x

⁢

g

1

⁢

(

y

)

∈

H

⁢

(

k

)

and

e

i

⁢

k

⁢

x

⁢

g

2

⁢

(

y

)

∈

H

⁢

(

k

)

as

∫

-

∞

+

∞

g

1

⁢

(

y

)

⁢

g

2

⁢

(

y

)

. (By the way, note that

H

⁢

(

k

)

is not a subset of

L

2

⁢

(

ℝ

2

)

because the functions contained in

H

⁢

(

k

)

are not square-integrable in the

x

variable

.)

By the

definition

, the space

∫

⊕

-

∞

+

∞

H

⁢

(

k

)

⁢

𝑑

k

consists of functions

(

k

,

x

,

y

)

↦

e

i

⁢

k

⁢

x

⁢

g

⁢

(

k

,

y

)

such that

∫

-

∞

+

∞

∥

g

⁢

(

k

)

∥

H

⁢

(

k

)

=

∫

-

∞

+

∞

∫

-

∞

+

∞

|

g

⁢

(

k

,

y

)

|

2

⁢

𝑑

y

⁢

𝑑

k

<

∞

Elements of this space may be regarded as functions of

x

and

y

in a natural way:

g

~

⁢

(

x

,

y

)

=

∫

-

∞

+

∞

e

i

⁢

k

⁢

x

⁢

g

⁢

(

k

,

y

)

By Parseval’s

identity

,

g

~

∈

L

2

⁢

(

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

H

⁢

(

k

)

in the example can be regarded as the “eigenspace” of the formally self-adjoint operator

-

i

⁢

∂

/

∂

⁡

x

with

eigenvalue

k

. (It is not eigenspaces in the strict sense of the term since its elements are not

square integrable

in

x

.) 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

x

axis. Each of the spaces

H

⁢

(

k

)

transforms under a different representation of this group — under translation by an amount

x

0

along the

x

axis, an element of

H

⁢

(

k

)

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

.

Title

direct integral of Hilbert spaces

Canonical

name

DirectIntegralOfHilbertSpaces

Date of creation

2013-03-22 14:43:57

Last modified on

2013-03-22 14:43:57

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 46C05

Defines

direct integral

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