spectral measure

spectral measure

1

Definition

In this entry by a

projection

we an orthogonal projection over some

Hilbert space

. Also, we say that two projections are

orthogonal

if their images are

orthogonal subspaces

.

Let

H

be an Hilbert space,

B

⁢

(

H

)

the

algebra

of

bounded operators

in

H

and

(

X

,

ℬ

)

a

measurable space

.

Definition

-

A

spectral measure

in

X

is a

function

P

:

B

⟶

B

⁢

(

H

)

such that

a)

P

⁢

(

E

)

is a projection in

B

⁢

(

H

)

for every

E

∈

B

.

b)

P

⁢

(

∅

)

=

0

.

c)

P

⁢

(

X

)

=

I

, where

I

denotes the

identity operator

in

B

⁢

(

H

)

.

d)

If

E

1

and

E

2

are disjoint subsets of

B

, then

P

⁢

(

E

1

)

and

P

⁢

(

E

2

)

are orthogonal.

e)

P

⁢

(

⋃

n

=

1

∞

E

n

)

=

∑

n

=

1

∞

P

⁢

(

E

n

)

for every

sequence

E

1

,

E

2

,

…

of

disjoint sets

in

B

.

The in the last condition is interpreted as the

pointwise

limit of the

partial sums

. Since from condition (d) the projections

P

⁢

(

E

1

)

,

P

⁢

(

E

2

)

,

…

are orthogonal, we know that the pointwise limit exists and is a projection (see this entry (

http://planetmath.org/LatticeOfProjections

),

Theorem

5).

:

In the following

Ran

⁢

(

T

)

denotes the range (

http://planetmath.org/Function

) of an

operator

T

∈

B

⁢

(

H

)

.

•

E

1

⊆

E

2

⟹

Ran

⁢

(

P

⁢

(

E

1

)

)

⊆

Ran

⁢

(

P

⁢

(

E

2

)

)

.

•

P

⁢

(

E

1

∩

E

2

)

=

P

⁢

(

E

1

)

⁢

P

⁢

(

E

2

)

for every

E

1

,

E

2

∈

ℬ

.

Thus, a spectral measure is a

countably additive vector measure

whose values are projections. For that, spectral measures are also called

projection valued measures

.

2

Examples

•

Let

(

X

,

ℬ

,

μ

)

be a

measure space

. Consider the Hilbert space

L

2

⁢

(

X

,

μ

)

(

http://planetmath.org/L2SpacesAreHilbertSpaces

). We regard a function

f

in

L

∞

⁢

(

X

,

μ

)

(

http://planetmath.org/LpSpace

) as the

multiplication operator

M

f

∈

B

⁢

(

L

2

⁢

(

X

,

μ

)

)

given by

M

f

⁢

(

ξ

)

=

f

⁢

ξ

,

ξ

∈

L

2

⁢

(

X

,

μ

)

In this setting, the

characteristic functions

are projections in

B

⁢

(

L

2

⁢

(

X

,

μ

)

)

and we have a spectral measure given by

P

:

X

⟶

B

⁢

(

L

2

⁢

(

X

,

μ

)

)

P

⁢

(

E

)

:=

χ

E

•

Let

H

be a Hilbert space,

T

∈

B

⁢

(

H

)

a

normal operator

and

σ

⁢

(

T

)

the

spectrum

of

T

. For any measurable subset

E

⊆

σ

⁢

(

T

)

the operators

χ

E

⁢

(

T

)

, given by the

Borel functional calculus

, are projections in

B

⁢

(

H

)

. Moreover, we have a spectral measure given by:

P

:

X

⟶

B

⁢

(

H

)

P

⁢

(

E

)

:=

χ

E

⁢

(

T

)

3

Equivalent Definition

The following result provides a very useful

equivalent

definition of a spectral measure.

Theorem 1 -

A function

P

:

B

⟶

B

⁢

(

H

)

whose values are projections is a spectral measure in

X

if and only if

P

⁢

(

X

)

=

I

and for every

ξ

,

η

∈

H

the function

μ

ξ

,

η

:

X

⟶

C

given by

μ

ξ

,

η

⁢

(

E

)

:=

⟨

P

⁢

(

E

)

⁢

ξ

,

η

⟩

is a

complex measure

in

X

.

4

Integration against spectral measures

Let

f

:

X

⟶

ℂ

be a

bounded

(

http://planetmath.org/Bounded

)

measurable function

and

P

a spectral measure in

X

. We are interested to give meaning to the

integral

∫

X

f

⁢

𝑑

P

Since we are dealing with “

measures

” whose values are

linear operators

it is reasonable to expect that this integral is itself a linear operator.

There are two natural ways to define it that turn out to be equivalent. The first approach is a construction that resembles the approximation of

f

by

simple functions

in

Lebesgue integral

theory. Here the role of simple functions will be played by the operators of the form

∑

i

λ

i

⁢

P

⁢

(

E

i

)

,

λ

i

∈

ℂ

Theorem 2 -

There exists a unique operator

S

∈

B

⁢

(

H

)

with the following

property

: for any given

ϵ

>

0

and for every

measurable partition

{

E

1

,

⋯

,

E

n

}

of

X

that

satisfies

|

f

⁢

(

x

)

-

f

⁢

(

x

′

)

|

<

ϵ

for all

x

,

x

′

∈

E

i

, we have

∥

S

-

∑

i

=

1

n

f

⁢

(

x

i

)

⁢

P

⁢

(

E

i

)

∥

<

ϵ

for any choice of points

x

i

∈

E

i

.

We can then define

∫

X

f

⁢

𝑑

P

as the unique operator

S

described by Theorem 2.

The other approach to define this integral is by specifying an appropriate

bounded sesquilinear form

. Recall that from

Riesz representation theorem

(

http://planetmath.org/RieszRepresentationTheoremOfBoundedSesquilinearForms

), to every bounded sesquilinear form corresponds a unique bounded operator. The construction is as follows:

First we notice that, from the alternative defintion of spectral measure (Theorem 1), for every vectors

ξ

,

η

∈

H

we can define a complex measure

μ

ξ

,

η

by

μ

ξ

,

η

⁢

(

E

)

=

⟨

P

⁢

(

E

)

⁢

ξ

,

η

⟩

,

whose

total variation

is estimated by

∥

μ

ξ

,

η

∥

≤

∥

ξ

∥

⁢

∥

η

∥

.

Then we notice that the function

[

⋅

,

⋅

]

:

H

×

H

⟶

ℂ

defined by

[

ξ

,

η

]

:=

∫

X

f

⁢

𝑑

μ

ξ

,

η

is a

sesquilinear form

.

Then, by the Riesz representation theorem (

http://planetmath.org/RieszRepresentationTheoremOfBoundedSesquilinearForms

), there exists a unique operator

S

∈

B

⁢

(

H

)

such that

⟨

S

⁢

ξ

,

η

⟩

=

∫

X

f

⁢

𝑑

μ

ξ

,

η

,

ξ

,

η

∈

H

(1)

We can then define

∫

X

f

⁢

𝑑

P

as this operator

S

. Of course, the two definitions are equivalent. We summarize this in the following result

Theorem 3 -

Given a spectral measure

P

and a bounded

Borel function

f

, an operator

S

that satisfies condition (1) also satisfies the conditions of Theorem 2. Therefore, both definitions of the integral of

f

with respect to

P

coincide and we have that:

•

⟨

∫

X

f

⁢

𝑑

P

⁢

ξ

,

η

⟩

=

∫

X

f

⁢

𝑑

μ

ξ

,

η

•

∫

X

f

⁢

𝑑

P

can be arbitrarilly approximated in norm by operators of the form

∑

i

=

1

n

f

⁢

(

x

i

)

⁢

P

⁢

(

E

i

)

.

5

Remarks

The second example we gave above, of a spectral measure associated with a normal operator, is in some sense the general case: all spectral projections in

ℂ

supported in a

compact set

arise from a normal operator. Thus, to any such spectral projection we can associate a normal operator and vice-versa. This interplay between spectral projections and normal operators is deeply explored in some versions of the

spectral theorem

.

References

1

W. Arveson,

A Short Course on

Spectral Theory

, Graduate Texts in Mathematics, 209, Springer, New York, 2002

2

J. B. Conway,

A Course in

Functional Analysis

, 2nd ed., Graduate Texts in Mathematics, 96, Springer-Verlag, New York, Berlin, 1990.

Title

spectral measure

Canonical

name

SpectralMeasure

Date of creation

2013-03-22 17:32:06

Last modified on

2013-03-22 17:32:06

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

11

Author

asteroid (17536)

Entry type

Definition

Classification

msc 47A56

Classification

msc 46G12

Classification

msc 46G10

Classification

msc 28C20

Classification

msc 28B05

Synonym

projection valued measure

Defines

integration against spectral measures

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