invariant subspaces for self-adjoint *-algebras of operators

invariant subspaces for self-adjoint *-algebras of operators

In this entry we provide few results concerning

invariant subspaces

of *-algebras of

bounded operators

on

Hilbert spaces

.

Let

H

be a Hilbert space and

B

⁢

(

H

)

its algebra of bounded operators. Recall that, given an

operator

T

∈

B

⁢

(

H

)

, a

subspace

V

⊆

H

is said to be

invariant

for

T

if

T

⁢

x

∈

V

whenever

x

∈

V

.

Similarly, given a

subalgebra

𝒜

⊆

B

⁢

(

H

)

, we will say that a subspace

V

⊆

H

is

invariant

for

𝒜

if

T

⁢

x

∈

V

whenever

T

∈

𝒜

and

x

∈

V

, i.e. if

V

is invariant for all operators in

𝒜

.

Invariant subspaces for a single operator

Proposition

1 -

Let

T

∈

B

⁢

(

H

)

. If a subspace

V

⊂

H

is invariant for

T

, then so is its

closure

V

¯

.

Proof:

Let

x

∈

V

¯

. There is a

sequence

{

x

n

}

in

V

such that

x

n

→

x

. Hence,

T

⁢

x

n

→

T

⁢

x

. Since

V

is invariant for

T

, all

T

⁢

x

n

belong to

V

. Thus, their limit

T

⁢

x

must be in

V

¯

. We conclude that

V

¯

is also invariant for

T

.

□

Proposition 2 -

Let

T

∈

B

⁢

(

H

)

. If a subspace

V

⊂

H

is invariant for

T

, then its

orthogonal complement

V

⟂

is invariant for

T

*

.

Proof:

Let

y

∈

V

⟂

. For all

x

∈

H

we have that

⟨

x

,

T

*

⁢

y

⟩

=

⟨

T

⁢

x

,

y

⟩

=

0

, where the last

equality

comes from the fact that

T

⁢

x

∈

V

, since

V

is invariant for

T

. Therefore

T

*

⁢

y

must belong to

V

⟂

, from which we conclude that

V

⟂

is invariant for

T

*

.

□

Proposition 3 -

Let

T

∈

B

⁢

(

H

)

,

V

⊂

H

a closed subspace and

P

∈

B

⁢

(

H

)

the

orthogonal projection

onto

V

. The following are statements are

equivalent

:

1.

V

is invariant for

T

.

2.

V

⟂

is invariant for

T

*

.

3.

T

⁢

P

=

P

⁢

T

⁢

P

.

Proof:

(

1

)

⟹

(

2

)

This part follows directly from Proposition 2.

(

2

)

⟹

(

1

)

From Proposition 2 it follows that

(

V

⟂

)

⟂

is invariant for

(

T

*

)

*

=

T

. Since

V

is closed,

V

=

V

¯

=

(

V

⟂

)

⟂

. We conclude that

V

is invariant for

T

.

(

1

)

⟹

(

3

)

Let

x

∈

H

. From the

orthogonal decomposition theorem

we know that

H

=

V

⊕

V

⟂

, hence

x

=

y

+

z

, where

y

∈

V

and

z

∈

V

⟂

. We now see that

T

⁢

P

⁢

x

=

T

⁢

y

and

P

⁢

T

⁢

P

⁢

x

=

P

⁢

T

⁢

y

=

T

⁢

y

, where the last equality comes from the fact that

T

⁢

y

∈

V

. Hence,

T

⁢

P

=

P

⁢

T

⁢

P

.

(

3

)

⟹

(

1

)

Let

x

∈

V

. We have that

T

⁢

x

=

T

⁢

P

⁢

x

=

P

⁢

T

⁢

P

⁢

x

. Since

P

⁢

T

⁢

P

⁢

x

is obviously on the image of

P

, it follows that

T

⁢

x

∈

V

, i.e.

V

is invariant for

T

.

□

Proposition 4 -

Let

T

∈

B

⁢

(

H

)

,

V

⊂

H

a closed subspace and

P

∈

B

⁢

(

H

)

the orhtogonal projection onto

V

. The subspaces

V

and

V

⟂

are both invariant for

T

if and only if

T

⁢

P

=

P

⁢

T

.

Proof:

(

⟹

)

From Proposition 3 it follows that

V

is invariant for both

T

and

T

*

. Then, again from Proposition 3, we see that

P

⁢

T

=

(

T

*

⁢

P

)

*

=

(

P

⁢

T

*

⁢

P

)

*

=

P

⁢

T

⁢

P

=

T

⁢

P

.

(

⟸

)

Suppose

T

⁢

P

=

P

⁢

T

. Then

P

⁢

T

⁢

P

=

T

⁢

P

⁢

P

=

T

⁢

P

, and from Proposition 3 we see that

V

is invariant for

T

.

We also have that

P

⁢

T

*

=

T

*

⁢

P

, and we can conclude in the same way that

V

is invariant for

T

*

. From Proposition 3 it follows that

V

⟂

is also invariant for

T

.

□

Invariant subspaces for *-algebras of operators

We shall now generalize some of the above results to the case of

self-adjoint

subalgebras of

B

⁢

(

H

)

.

Proposition 5 -

Let

A

be a *-subalgebra of

B

⁢

(

H

)

and

V

a subspace of

H

. If a subspace

V

is invariant for

A

, then so are its closure

V

¯

and its orthogonal complement

V

⟂

.

Proof:

From Proposition 1 it follows that

V

¯

is invariant for all operators in

𝒜

, which means that

V

is invariant for

𝒜

.

Also, from Proposition 2 it follows that

V

⟂

is invariant for the

adjoint

of each operator in

𝒜

. Since

𝒜

is self-adjoint, it follows that

V

⟂

is invariant for

𝒜

.

□

Theorem

-

Let

A

be a *-subalgebra of

B

⁢

(

H

)

,

V

⊂

H

a closed subspace and

P

the orthogonal projection onto

V

. The following are equivalent:

1.

V

is invariant for

A

.

2.

V

⟂

is invariant for

A

.

3.

P

∈

A

′

, i.e.

P

belongs to the

commutant

of

A

.

Proof:

(

1

)

⟺

(

2

)

This equivalence follows directly from Proposition 5 and the fact that

V

is closed.

(

1

)

⟹

(

3

)

Suppose

V

is invariant for

𝒜

. We have already proved that

V

⟂

is also invariant for

𝒜

. Thus, from Proposition 4 it follows that

P

commutes with all operators in

𝒜

, i.e.

P

∈

𝒜

′

.

(

3

)

⟹

(

1

)

Suppose

P

∈

𝒜

′

. Then

P

commutes with all operators in

𝒜

. From Proposition 4 it follows that

V

is invariant for each operator in

𝒜

, i.e.

V

is invariant for

𝒜

.

□

Title

invariant subspaces for self-adjoint *-algebras of operators

Canonical name

InvariantSubspacesForSelfadjointalgebrasOfOperators

Date of creation

2013-03-22 18:40:23

Last modified on

2013-03-22 18:40:23

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

9

Author

asteroid (17536)

Entry type

Feature

Classification

msc 46K05

Classification

msc 46H35

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