proof of Borel functional calculus

proof of Borel functional calculus

In this entry we give a proof of the main result about the

Borel functional calculus

( 1 on the parent entry). We will restate here the result for convenience. Please, check the parent entry for the details on notation.

Theorem

-

Let

T

be a

normal operator

in

B

⁢

(

H

)

and

π

:

C

⁢

(

σ

⁢

(

T

)

)

⟶

B

⁢

(

H

)

the unital *-

homomorphism

corresponding to the

continuous functional calculus

for

T

. Then,

π

extends uniquely to a *-homomorphism

π

~

:

B

⁢

(

σ

⁢

(

T

)

)

⟶

B

⁢

(

H

)

that is

continuous

from the

μ

-

topology

to the

weak operator topology

. Moreover, each

operator

π

⁢

(

f

)

lies in strong operator (

http://planetmath.org/OperatorTopologies

)

closure

of the unital *-algebra

generated by

T

.

Proof :

First we shall prove the

existence

of the

extension

(

http://planetmath.org/ExtensionOfAFunction

)

π

~

with the described

continuity

property, then we shall prove its uniqueness and for last we shall prove the last assertion of the theorem about the image of

π

~

. For simplicity the proofs of some auxiliary results are given at the end of the entry

0.0.1

Existence

For each pair of vectors

ξ

,

η

∈

H

consider the

linear functional

ϕ

ξ

,

η

:

C

⁢

(

σ

⁢

(

T

)

)

→

ℂ

given by

ϕ

ξ

,

η

⁢

(

f

)

:=

⟨

π

⁢

(

f

)

⁢

ξ

,

η

⟩

This linear functional is

bounded

with norm at most

∥

ξ

∥

⁢

∥

η

∥

because

∥

ϕ

ξ

,

η

⁢

(

f

)

∥

≤

∥

π

∥

⁢

∥

f

∥

⁢

∥

ξ

∥

⁢

∥

η

∥

=

∥

f

∥

⁢

∥

ξ

∥

⁢

∥

η

∥

where the last

equality

comes from the fact that

π

is a *-

isomorphism

between

C

*

-algebras (

http://planetmath.org/CAlgebras

), therefore having norm 1 (see this entry (

http://planetmath.org/HomomorphismsOfCAlgebrasAreContinuous

)).

By the

Riesz representation theorem

(

http://planetmath.org/RieszRepresentationTheoremOfLinearFunctionalsOnFunctionSpaces

) there is a unique complex

Radon measure

μ

ξ

,

η

in

σ

⁢

(

T

)

such that

⟨

π

⁢

(

f

)

⁢

ξ

,

η

⟩

=

∫

σ

⁢

(

T

)

f

⁢

𝑑

μ

ξ

,

η

,

∀

f

∈

C

⁢

(

σ

⁢

(

T

)

)

(1)

Consider now the

mapping

(

ξ

,

η

)

↦

μ

ξ

,

η

. This mapping has the following properties, whoose proofs are given at the end of the entry (

section

Auxiliary Results):

a)

μ

λ

1

⁢

ξ

1

+

λ

2

⁢

ξ

2

,

η

=

λ

1

⁢

μ

ξ

1

,

η

+

λ

2

⁢

μ

ξ

2

,

η

, for all

λ

1

,

λ

2

∈

ℂ

.

b)

μ

ξ

,

λ

1

⁢

η

1

+

λ

2

⁢

η

2

=

λ

1

¯

⁢

μ

ξ

,

η

1

+

λ

2

¯

⁢

μ

ξ

,

η

2

, for all

λ

1

,

λ

2

∈

ℂ

.

c)

μ

ξ

,

η

=

μ

η

,

ξ

¯

.

d)

g

⋅

μ

ξ

,

η

=

μ

π

⁢

(

g

)

⁢

ξ

,

η

, for all

g

∈

C

(

σ

(

T

)

.

Therefore, for each

function

f

∈

B

⁢

(

σ

⁢

(

T

)

)

we have a

sesquilinear form

in

H

given by

[

ξ

,

η

]

:=

∫

σ

⁢

(

T

)

f

⁢

𝑑

μ

ξ

,

η

Moreover, this sesquilinear form is bounded (

http://planetmath.org/BoundedSesquilinearForm

) with norm at most

∥

f

∥

⁢

∥

ξ

∥

⁢

∥

η

∥

. By the

Riesz lemma on bounded sesquilinear forms

, there is a unique operator

π

~

⁢

(

f

)

∈

B

⁢

(

H

)

such that

⟨

π

~

⁢

(

f

)

⁢

ξ

,

η

⟩

:=

∫

σ

⁢

(

T

)

f

⁢

𝑑

μ

ξ

,

η

,

ξ

,

η

∈

H

We will now see that the mapping

π

~

:

B

(

σ

(

T

)

→

B

(

H

)

, such that

f

↦

π

~

⁢

(

f

)

, has the desired properties stated in the theorem.

First, it is clear that

π

~

is linear. Also clear is the fact that

π

~

coincides with

π

in

C

⁢

(

σ

⁢

(

T

)

)

, because of equality (1) and the uniqueness part of Riesz lemma on sesquilinear forms. Now, for any real valued function

f

∈

B

⁢

(

σ

⁢

(

T

)

)

we have that

⟨

π

~

⁢

(

f

)

*

⁢

ξ

,

η

⟩

=

⟨

π

~

⁢

(

f

)

⁢

η

,

ξ

⟩

¯

=

∫

σ

⁢

(

T

)

f

⁢

𝑑

μ

η

,

ξ

¯

=

∫

σ

⁢

(

T

)

f

⁢

𝑑

μ

ξ

,

η

=

⟨

π

⁢

(

f

)

⁢

ξ

,

η

⟩

which means that

π

~

⁢

(

f

)

*

=

π

~

⁢

(

f

)

, i.e.

π

~

⁢

(

f

)

is

self-adjoint

. Decomposing an arbitrary function

f

∈

B

⁢

(

σ

⁢

(

T

)

)

in its real and imaginary parts we see that

π

~

⁢

(

f

)

*

=

π

~

⁢

(

f

¯

)

.

We now show that

π

~

is multiplicative, i.e.

π

~

⁢

(

f

⁢

g

)

=

π

~

⁢

(

f

)

⁢

π

~

⁢

(

g

)

for all

f

,

g

∈

B

⁢

(

σ

⁢

(

T

)

)

. For that we need an additional property of the

measures

μ

ξ

,

η

, whoose proof is also at the end of the entry:

e)

f

⋅

μ

ξ

,

η

=

μ

ξ

,

π

~

⁢

(

f

)

*

⁢

η

, for all

f

∈

B

⁢

(

σ

⁢

(

T

)

)

.

Given

f

,

g

∈

B

⁢

(

σ

⁢

(

T

)

)

we have, for every

ξ

,

η

∈

H

,

⟨

π

~

⁢

(

f

⁢

g

)

⁢

ξ

,

η

⟩

=

∫

σ

⁢

(

T

)

f

⁢

g

⁢

𝑑

μ

ξ

,

η

=

∫

σ

(

T

)

g

⁢

𝑑

μ

ξ

,

π

~

⁢

(

f

)

*

η

=

⟨

π

~

⁢

(

g

)

⁢

ξ

,

π

~

⁢

(

f

)

*

⁢

η

⟩

=

⟨

π

~

⁢

(

f

)

⁢

π

~

⁢

(

g

)

⁢

ξ

,

η

⟩

and therefore

π

~

⁢

(

f

⁢

g

)

=

π

~

⁢

(

f

)

⁢

π

~

⁢

(

g

)

. Thus,

π

~

:

B

⁢

(

σ

⁢

(

T

)

)

→

B

⁢

(

H

)

is a *-homomorphism that extends

π

.

0.0.2

Continuity Property

We now prove that the above defined

π

~

is continuous from the

μ

-topology to the weak operator topology.

Let

{

f

i

}

be a net of functions in

B

⁢

(

σ

⁢

(

T

)

)

that

converge

in the

μ

-topology to a function

f

∈

B

⁢

(

σ

⁢

(

T

)

)

. This means that for all Radon measures

ν

in

σ

⁢

(

T

)

we have

∫

f

i

⁢

𝑑

ν

→

∫

f

⁢

𝑑

ν

.

Now for all

ξ

,

η

∈

H

we have

|

⟨

π

~

⁢

(

f

i

-

f

)

⁢

ξ

,

η

⟩

|

=

|

∫

σ

⁢

(

T

)

f

i

-

f

⁢

d

⁢

μ

ξ

,

η

|

→

0

Hence,

π

~

⁢

(

f

i

)

converges to

π

~

⁢

(

f

)

in the weak operator topology.

0.0.3

Uniqueness

Let

π

′

:

B

⁢

(

σ

⁢

(

T

)

)

→

B

⁢

(

H

)

be another *-homomorphism that extends

π

and is continuous from the

μ

-topology to the weak operator topology. For any measurable subset

S

⊂

σ

⁢

(

T

)

consider the set

W

:=

{

(

U

,

K

)

:

U

⊃

S

⁢

is open and

⁢

K

⊂

S

⁢

is compact

}

We give this set the

partial order

≤

such that

(

U

1

,

K

1

)

≤

(

U

2

,

K

2

)

whenever

U

2

⊂

U

1

and

K

1

⊂

K

2

. For any pair

(

U

,

K

)

∈

W

there is a

continuous function

f

U

,

K

∈

C

⁢

(

σ

⁢

(

T

)

)

such that

f

takes values on the

interval

[

0

,

1

]

,

f

|

K

=

1

and

supp

⁢

f

⊂

U

(see this entry (

http://planetmath.org/ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces

)).

We claim that

f

U

,

K

converges to

χ

S

in the

μ

-topology. In fact, given a complex Radon measure

ν

in

σ

⁢

(

T

)

, there is for every

ϵ

>

0

a pair

(

U

0

,

K

0

)

∈

W

such that

|

ν

|

⁢

(

U

0

∖

K

0

)

<

ϵ

. Of course, for all pairs

(

U

,

K

)

such that

(

U

0

,

K

0

)

≤

(

U

,

K

)

we also have

|

ν

|

⁢

(

U

∖

K

)

<

ϵ

. Hence, we have

|

∫

σ

⁢

(

T

)

f

U

,

K

-

χ

S

⁢

d

⁢

ν

|

≤

∫

U

∖

K

|

f

U

,

K

-

χ

S

|

⁢

d

⁢

|

ν

|

≤

ϵ

We conclude that

f

U

,

K

converges to

χ

S

in the

μ

-topology.

Since

π

′

is continuous from the

μ

-topology to the weak operator topology we must have

⟨

π

′

⁢

(

f

U

,

V

)

⁢

ξ

,

η

⟩

⟶

⟨

π

′

⁢

(

χ

S

)

⁢

ξ

,

η

⟩

But since

π

′

and

π

~

coincide with

π

on

C

⁢

(

σ

⁢

(

T

)

)

we also have

⟨

π

′

⁢

(

f

U

,

V

)

⁢

ξ

,

η

⟩

=

⟨

π

~

⁢

(

f

U

,

V

)

⁢

ξ

,

η

⟩

⟶

⟨

π

~

⁢

(

χ

S

)

⁢

ξ

,

η

⟩

Hence, for any

characteristic function

χ

S

we have

π

′

⁢

(

χ

S

)

=

π

~

⁢

(

χ

S

)

. Since any function

f

∈

B

⁢

(

σ

⁢

(

T

)

)

can be uniformly approximated by

simple functions

it follows that

π

′

⁢

(

f

)

=

π

~

⁢

(

f

)

, and we have proved the uniqueness of

π

~

.

0.0.4

Image of

π

~

Let

𝒜

be the unital *-algebra generated by

T

. We now prove that for any

f

∈

B

⁢

(

σ

⁢

(

T

)

)

, the operator

π

~

⁢

(

f

)

lies in the strong operator closure of

𝒜

, i.e. lies in the

von Neumann algebra

generated by

T

. For that it is enough to prove that

π

~

⁢

(

f

)

is in the

double commutant

𝒜

′′

of

𝒜

.

Recall from the continuous functional calculus that

π

⁢

(

f

)

is in the norm closure of

𝒜

, and hence in

𝒜

′′

, for every

f

∈

C

⁢

(

σ

⁢

(

T

)

)

.

We have seen above that for each characteristic function

χ

S

there is a net

f

U

,

K

of functions in

C

⁢

(

σ

⁢

(

T

)

)

such that

π

~

⁢

(

f

U

,

K

)

→

π

~

⁢

(

χ

S

)

in the weak operator topology. Given an

element

R

in the commutant of

𝒜

we have

⟨

π

~

⁢

(

f

U

,

K

)

⁢

R

⁢

ξ

,

η

⟩

=

⟨

R

⁢

π

~

⁢

(

f

U

,

K

)

⁢

ξ

,

η

⟩

,

∀

ξ

,

η

∈

H

The first term converges to

⟨

π

~

⁢

(

χ

S

)

⁢

R

⁢

ξ

,

η

⟩

, whereas the second to

⟨

R

⁢

π

~

⁢

(

χ

S

)

⁢

ξ

,

η

⟩

. Thus,

π

~

⁢

(

χ

S

)

⁢

R

=

R

⁢

π

~

⁢

(

χ

S

)

, and therefore

π

~

⁢

(

χ

S

)

∈

𝒜

′′

.

Since every function

f

∈

B

⁢

(

σ

⁢

(

T

)

)

can be uniformly approximated by simple functions, it follows that

π

~

⁢

(

f

)

∈

𝒜

′′

.

0.0.5

Auxiliary Results

In this section we prove the properties of the measures

μ

ξ

,

η

stated and used above.

a)

For all functions

f

∈

C

⁢

(

σ

⁢

(

T

)

)

we have

⟨

π

⁢

(

f

)

⁢

(

λ

1

⁢

ξ

1

+

λ

2

⁢

ξ

2

)

,

η

⟩

=

λ

1

⁢

⟨

π

⁢

(

f

)

⁢

ξ

1

,

η

⟩

+

λ

2

⁢

⟨

π

⁢

(

f

)

⁢

ξ

2

,

η

⟩

. Hence,

∫

σ

⁢

(

T

)

f

⁢

𝑑

μ

λ

1

⁢

ξ

1

+

λ

2

⁢

ξ

2

,

η

=

λ

1

⁢

∫

σ

⁢

(

T

)

f

⁢

𝑑

μ

ξ

1

,

η

+

λ

2

⁢

∫

σ

⁢

(

T

)

f

⁢

𝑑

μ

ξ

2

,

η

Since this holds for every

f

∈

C

⁢

(

σ

⁢

(

T

)

)

, the uniqueness part of tells us that

μ

λ

1

⁢

ξ

1

+

λ

2

⁢

ξ

2

,

η

=

λ

1

⁢

μ

ξ

1

,

η

+

λ

2

⁢

μ

ξ

2

,

η

b)

The proof is

similar

to a).

c)

For every

f

∈

C

⁢

(

σ

⁢

(

T

)

)

we have

∫

σ

⁢

(

T

)

f

⁢

𝑑

μ

ξ

,

η

=

⟨

π

⁢

(

f

)

⁢

ξ

,

η

⟩

=

⟨

π

⁢

(

f

¯

)

⁢

η

,

ξ

⟩

¯

=

∫

σ

⁢

(

T

)

f

¯

⁢

𝑑

μ

η

,

ξ

¯

=

∫

σ

⁢

(

T

)

f

⁢

𝑑

μ

η

,

ξ

¯

Hence we conclude that

μ

x

⁢

i

,

η

=

μ

η

,

ξ

¯

.

d)

For every

f

∈

C

⁢

(

σ

⁢

(

T

)

)

we have

∫

σ

⁢

(

T

)

f

⁢

𝑑

g

⋅

μ

ξ

,

η

=

∫

σ

⁢

(

T

)

f

⁢

g

⁢

𝑑

μ

ξ

,

η

=

⟨

π

⁢

(

f

⁢

g

)

⁢

ξ

,

η

⟩

=

⟨

π

⁢

(

f

)

⁢

π

⁢

(

g

)

⁢

ξ

,

η

⟩

=

∫

σ

⁢

(

T

)

f

⁢

𝑑

μ

π

⁢

(

g

)

⁢

ξ

,

η

Hence,

g

⋅

μ

ξ

,

η

=

μ

π

⁢

(

g

)

⁢

ξ

,

η

.

e)

For all

h

∈

C

⁢

(

σ

⁢

(

T

)

)

we have

∫

σ

⁢

(

T

)

h

⁢

𝑑

f

⋅

μ

ξ

,

η

=

∫

σ

⁢

(

T

)

h

⁢

f

⁢

𝑑

μ

ξ

,

η

=

∫

σ

⁢

(

T

)

f

⁢

𝑑

h

⋅

μ

ξ

,

η

=

∫

σ

⁢

(

T

)

f

⁢

𝑑

μ

π

⁢

(

f

)

⁢

ξ

,

η

=

⟨

π

~

⁢

(

f

)

⁢

π

⁢

(

g

)

⁢

ξ

,

η

⟩

=

⟨

π

⁢

(

h

)

⁢

ξ

,

π

~

⁢

(

f

)

*

⁢

η

⟩

=

∫

σ

⁢

(

T

)

h

⁢

𝑑

μ

ξ

,

π

~

⁢

(

f

)

*

⁢

η

Hence,

f

⋅

μ

ξ

,

η

=

μ

ξ

,

π

~

⁢

(

f

)

*

⁢

η

.

Title

proof of Borel functional calculus

Canonical name

ProofOfBorelFunctionalCalculus

Date of creation

2013-03-22 18:50:26

Last modified on

2013-03-22 18:50:26

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

8

Author

asteroid (17536)

Entry type

Proof

Classification

msc 47A60

Classification

msc 46L10

Classification

msc 46H30

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