operator topologies

operator topologies

Let

X

be a

normed vector space

and

B

⁢

(

X

)

the space of

bounded operators

in

X

. There are several interesting

topologies

that can be given to

B

⁢

(

X

)

. In what follows,

T

α

denotes a net in

B

⁢

(

X

)

and

T

denotes an element of

B

⁢

(

X

)

.

Note: On 4, 5, 6 and 7,

X

must be a

Hilbert space

.

0.1

1. Norm Topology

This is the topology

induced

by the usual

operator norm

.

T

α

⟶

T

⁢

in the norm topology

⟺

∥

T

α

-

T

∥

⟶

0

0.2

2. Strong Operator Topology

This is the topology generated by the family of

semi-norms

∥

⋅

∥

x

,

x

∈

X

defined by

∥

T

∥

x

:=

∥

T

⁢

x

∥

. That means

T

α

⟶

T

in the strong operator topology

⟺

∥

(

T

α

-

T

)

x

∥

⟶

0

,

∀

x

∈

X

0.3

3. Weak Operator Topology

This is the topology generated by the family of semi-norms

∥

⋅

∥

f

,

x

, where

x

∈

X

and

f

is a

linear functional

of

X

(written

f

∈

X

*

, the

dual vector space

of

X

), defined by

∥

T

∥

f

,

x

:=

|

f

⁢

(

T

⁢

x

)

|

. That means

T

α

⟶

T

in the weak operator topology

⟺

∥

f

(

(

T

α

-

T

)

x

)

∥

⟶

0

,

∀

x

∈

X

,

∀

f

∈

X

*

In case

X

is an Hilbert space with

inner product

⟨

⋅

,

⋅

⟩

, we have that

T

α

⟶

T

in the weak operator topology

⟺

|

⟨

(

T

α

-

T

)

x

,

y

⟩

|

⟶

0

,

∀

x

,

y

∈

X

0.4

4.

σ

-Strong Operator Topology

In this topology

X

must be a Hilbert space. Let

K

⁢

(

X

)

denote the space of

compact operators

on

X

.

The

σ

-

strong operator topology

is the topology generated by the family of semi-norms

∥

⋅

∥

S

,

S

∈

K

(

X

)

, defined by

∥

T

∥

S

:=

∥

T

⁢

S

∥

. That means

T

α

⟶

T

in the

σ

-strong operator topology

⟺

∥

(

T

α

-

T

)

S

∥

⟶

0

,

∀

S

∈

K

(

X

)

Equivalently,

T

α

⟶

T

⟺

T

α

⁢

S

⟶

T

⁢

S

in norm for every

S

∈

K

⁢

(

X

)

.

This topology is also called the

ultra-strong operator topology

.

0.5

5.

σ

-Weak Operator Topology

In this topology

X

must be a Hilbert space. Let

B

⁢

(

X

)

*

denote the space of

trace-class operators

on

X

and

T

⁢

r

⁢

(

S

)

the trace of an

operator

S

∈

B

⁢

(

X

)

*

.

The

σ

-weak operator topology

is the topology generated by the family of semi-norms

{

ω

S

:

S

∈

B

⁢

(

X

)

*

}

defined by

ω

S

⁢

(

T

)

:=

|

T

⁢

r

⁢

(

T

⁢

S

)

|

. That means

T

α

⟶

T

in the

σ

-weak operator topology

⟺

|

T

r

[

(

T

α

-

T

)

S

]

|

⟶

0

,

∀

S

∈

B

(

X

)

*

This topology is also called the

ultra-weak operator topology

.

0.6

6. Strong-* Operator Topology

In this topology

X

must be a Hilbert space. In the following

T

*

denotes the

adjoint operator

of

T

.

The

strong-* operator topology

is the topology generated by the family of semi-norms

∥

⋅

∥

x

,

x

∈

X

defined by

∥

T

∥

x

:=

∥

T

⁢

x

∥

+

∥

T

*

⁢

x

∥

. That means

T

α

⟶

T

in the strong-* operator topology

⟺

∥

(

T

α

-

T

)

x

∥

+

∥

(

T

α

*

-

T

*

)

x

∥

⟶

0

,

∀

x

∈

X

Equivalently,

T

α

⟶

T

if and only if

T

α

⁢

x

⟶

T

⁢

x

and

T

α

*

⁢

x

⟶

T

*

⁢

x

, for every

x

∈

X

.

0.7

7.

σ

-Strong-* Operator Topology

In this topology

X

must be a Hilbert space. Let

K

⁢

(

X

)

denote the space of compact operators on

X

. In the following

T

*

denotes the adjoint operator of

T

.

The

σ

-strong-* operator topology

is the topology generated by the family of semi-norms

∥

⋅

∥

S

,

S

∈

K

(

X

)

defined by

∥

T

∥

S

:=

∥

T

⁢

S

∥

+

∥

T

*

⁢

S

∥

. That means

T

α

⟶

T

in the

σ

-strong-* operator topology

⟺

∥

(

T

α

-

T

)

S

∥

+

∥

(

T

α

*

-

T

*

)

S

∥

⟶

0

,

∀

S

∈

K

(

X

)

Equivalently,

T

α

⟶

T

if and only if

T

α

⁢

S

⟶

T

⁢

S

and

T

α

*

⁢

S

⟶

T

*

⁢

S

in norm, for every

S

∈

K

⁢

(

X

)

.

This topology is also called

ultra-strong-* operator topology

.

0.8

Comparison of Operator Topologies

•

The norm topology is the strongest of the topologies defined above.

•

The weak operator topology is weaker than the strong operator topology, which is weaker than the norm topology.

•

In Hilbert spaces we can summarize the relations of the above topologies in the following diagram. Given two topologies

𝒰

,

𝒱

the notation

𝒰

→

𝒱

means

𝒰

is weaker than

𝒱

:

\xymatrix

⁢

w

⁢

e

⁢

a

⁢

k

⁢

\ar

⁢

[

r

]

⁢

\ar

⁢

[

d

]

⁢

&

⁢

s

⁢

t

⁢

r

⁢

o

⁢

n

⁢

g

⁢

\ar

⁢

[

r

]

⁢

\ar

⁢

[

d

]

⁢

&

⁢

strong-*

\ar

⁢

[

d

]

⁢

σ

-weak

\ar

⁢

[

r

]

⁢

&

⁢

σ

-strong

\ar

⁢

[

r

]

⁢

&

⁢

σ

-strong-*

\ar

⁢

[

r

]

⁢

&

⁢

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⁢

o

⁢

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⁢

m

Title

operator topologies

Canonical

name

OperatorTopologies

Date of creation

2013-03-22 17:22:04

Last modified on

2013-03-22 17:22:04

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

18

Author

asteroid (17536)

Entry type

Definition

Classification

msc 54E99

Classification

msc 47L05

Classification

msc 46A32

Related topic

OperatorNorm

Defines

strong operator topology

Defines

weak operator topology

Defines

σ

-weak operator topology

Defines

σ

-strong operator topology

Defines

strong-* operator topology

Defines

σ

-strong-* operator topology

Defines

ultra-strong operator topology

Defines

ultra-weak operator topology

Defines

ultra-stro

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