aarjaneiro/planetmath huggingface dataset:
Gelfand transform

Gelfand transform

The Gelfand Transform

Let

𝒜

be a

Banach algebra

over

ℂ

.
Let

△

be the space of all

multiplicative linear functionals

in

𝒜

, endowed with the weak-*

topology

. Let

C

⁢

(

△

)

denote the

algebra

of complex valued

continuous functions

in

△

.

The

Gelfand transform

is the

mapping

^

:

𝒜

⟶

C

⁢

(

△

)

x

⟼

x

^

where

x

^

∈

C

⁢

(

△

)

is defined by

x

^

⁢

(

ϕ

)

:=

ϕ

⁢

(

x

)

,

∀

ϕ

∈

△

The Gelfand transform is a

continuous

homomorphism

from

𝒜

to

C

⁢

(

△

)

.

Theorem

-

Let

C

0

⁢

(

△

)

denote the algebra of complex valued continuous functions in

△

, that

vanish at infinity

.
The image of the Gelfand transform is

contained

in

C

0

⁢

(

△

)

.

The Gelfand transform is a very useful tool in the study of commutative Banach algebras and, particularly,

commutative

C

*

-algebras (

http://planetmath.org/CAlgebra

).

Classification of commutative

C

*

-algebras: Gelfand-Naimark theorems

The following results are called the

Gelfand-Naimark theorems

. They classify all commutative

C

*

-algebras and all commutative

C

*

-algebras with

identity element

.

Theorem 1 -

Let

𝒜

be a

C

*

-algebra over

ℂ

. Then

𝒜

is
*-

isomorphic

to

C

0

⁢

(

X

)

for some

locally compact Hausdorff space

X

. Moreover, the Gelfand transform is a
*-isomorphism between

𝒜

and

C

0

⁢

(

△

)

.

Theorem 2 -

Let

𝒜

be a unital

C

*

-algebra over

ℂ

. Then

𝒜

is
*-isomorphic to

C

⁢

(

X

)

for some

compact

Hausdorff space

X

. Moreover, the Gelfand transform is a
*-isomorphism between

𝒜

and

C

⁢

(

△

)

.

The above theorems can be substantially improved. In fact, there is an

equivalence

(

http://planetmath.org/EquivalenceOfCategories

) between the category of commutative

C

*

-algebras and the category of locally compact Hausdorff spaces. For more and details about this, see the entry about the general Gelfand-Naimark theorem (

http://planetmath.org/GelfandNaimarkTheorem

).

Title

Gelfand transform

Canonical name

GelfandTransform

Date of creation

2013-03-22 17:22:39

Last modified on

2013-03-22 17:22:39

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

26

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46L35

Classification

msc 46L05

Classification

msc 46J40

Classification

msc 46J05

Classification

msc 46H05

Related topic

MultiplicativeLinearFunctional

Related topic

NoncommutativeTopology

Related topic

CAlgebra3

Related topic

CAlgebra

Related topic

CompactQuantumGroup

Defines

classification of commutative

C

*

-algebras

Defines

commutative

C

*

-algebras classification

Defines

Gelfand-Naimark theorem

Generated on Fri Feb 9 21:44:15 2018 by

LaTeXML
Gelfand-Naimark theorem

Gelfand-Naimark theorem

Let

𝐇𝐚𝐮𝐬

be the

category

of

locally compact Hausdorff spaces

with

continuous proper maps

as

morphisms

.
And, let

𝐂

*

⁢

𝐀𝐥𝐠

be the category of

commutative

C

*

-

algebras

with proper *-

homomorphisms

(send

approximate units

into approximate units)
as morphisms.
There is a

contravariant functor

C

:

𝐇𝐚𝐮𝐬

op

→

𝐂

*

⁢

𝐀𝐥𝐠

which sends each locally compact Hausdorff space

X

to the commutative

C

*

-algebra

C

0

⁢

(

X

)

(

C

⁢

(

X

)

if

X

is

compact

).
Conversely, there is a contravariant functor

M

:

𝐂

*

⁢

𝐀𝐥𝐠

op

→

𝐇𝐚𝐮𝐬

which sends each commutative

C

*

-algebra

A

to the space of

characters

on

A

(with the Gelfand

topology

).

The

functors

C

and

M

are an

equivalence of categories

.

Title

Gelfand-Naimark theorem

Canonical name

GelfandNaimarkTheorem

Date of creation

2013-03-22 13:29:28

Last modified on

2013-03-22 13:29:28

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

5

Author

mhale (572)

Entry type

Theorem

Classification

msc 46L85

Generated on Fri Feb 9 21:44:19 2018 by

LaTeXML
Serre-Swan theorem

Serre-Swan theorem

Let

X

be a

compact

Hausdorff space

.
Let

𝐕𝐞𝐜

⁢

(

X

)

be the

category

of

complex vector bundles

over

X

.
And, let

𝐏𝐫𝐨𝐣𝐌𝐨𝐝

⁢

(

C

⁢

(

X

)

)

be the category of

finitely generated projective modules

over the

C

*

-

algebra

C

⁢

(

X

)

.
There is a

functor

Γ

:

𝐕𝐞𝐜

⁢

(

X

)

→

𝐏𝐫𝐨𝐣𝐌𝐨𝐝

⁢

(

C

⁢

(

X

)

)

which sends each complex vector bundle

E

→

X

to the

C

⁢

(

X

)

-module

Γ

⁢

(

X

,

E

)

of continuous sections.

The functor

Γ

is an

equivalence of categories

.

Title

Serre-Swan theorem

Canonical name

SerreSwanTheorem

Date of creation

2013-03-22 13:29:31

Last modified on

2013-03-22 13:29:31

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

5

Author

mhale (572)

Entry type

Theorem

Classification

msc 46L85

Generated on Fri Feb 9 21:44:21 2018 by

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locally compact groupoids

locally compact groupoids

1

Locally compact groupoids

This is a specific topic entry defining the basics of

locally compact groupoids

and related

concepts

.

Let us first recall the related concepts of

groupoid

and

topological groupoid

,
together with the appropriate notations needed to define a

locally compact groupoid

.

1.0.1

Groupoids and topological groupoids: categorical definitions

Recall that a groupoid

𝖦

is a

small category

with

inverses

over its set of objects

X

=

O

⁢

b

⁢

(

𝖦

)

. One writes

𝖦

x

y

for
the set of

morphisms

in

𝖦

from

x

to

y

.

A topological groupoid

consists of a space

𝖦

, a distinguished subspace

𝖦

(

0

)

=

Ob

⁢

(

𝖦

)

⊂

𝖦

, called

the space of objects

of

𝖦

,
together with maps

r

,

s

:

(1.1)

Generated on Fri Feb 9 21:44:25 2018 by

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Schwarzian derivative

Schwarzian derivative

If

g

is a thrice

differentiable function

, the

Schwarzian derivative

is defined by

S

⁢

(

g

)

=

2

⁢

g

′

⁢

g

′′′

-

3

⁢

(

g

′′

)

2

2

⁢

(

g

′

)

2

.

Title

Schwarzian derivative

Canonical name

SchwarzianDerivative

Date of creation

2013-03-22 17:17:14

Last modified on

2013-03-22 17:17:14

Owner

alexandros_plus (16765)

Last modified by

alexandros_plus (16765)

Numerical id

11

Author

alexandros_plus (16765)

Entry type

Definition

Classification

msc 46N99

Generated on Fri Feb 9 21:44:27 2018 by

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adjoint

adjoint

Let

ℋ

be a

Hilbert space

and let

A

:

𝒟

⁢

(

A

)

⊂

ℋ

→

ℋ

be a

densely defined

linear operator

. Suppose that for some

y

∈

ℋ

, there exists

z

∈

ℋ

such that

(

A

⁢

x

,

y

)

=

(

x

,

z

)

for all

x

∈

𝒟

⁢

(

A

)

. Then such

z

is unique, for if

z

′

is another element of

ℋ

satisfying that condition, we have

(

x

,

z

-

z

′

)

=

0

for all

x

∈

𝒟

⁢

(

A

)

, which implies

z

-

z

′

=

0

since

𝒟

⁢

(

A

)

is dense (

http://planetmath.org/Dense

). Hence we may define a new

operator

A

*

:

𝒟

⁢

(

A

*

)

⊂

ℋ

→

ℋ

by

𝒟

⁢

(

A

*

)

=

{

y

∈

ℋ

:

there is

z

∈

ℋ

such that

(

A

x

,

y

)

=

(

x

,

z

)

}

,

A

*

⁢

(

y

)

=

z

.

It is easy to see that

A

*

is linear, and it is called the

adjoint

of

A

.

Remark.

The requirement for

A

to be densely defined is essential, for otherwise we cannot guarantee

A

*

to be well defined.

Title

adjoint

Canonical name

Adjoint

Date of creation

2013-03-22 13:48:09

Last modified on

2013-03-22 13:48:09

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

10

Author

Koro (127)

Entry type

Definition

Classification

msc 47A05

Synonym

adjoint operator

Related topic

TransposeOperator

Generated on Fri Feb 9 21:44:49 2018 by

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Baker-Campbell-Hausdorff formula(e)

Baker-Campbell-Hausdorff formula(e)

Given a

linear operator

A

, we define:

exp

⁡

A

:=

∑

k

=

0

∞

1

k

!

⁢

A

k

.

(1)

It follows that

∂

∂

⁡

τ

⁢

e

τ

⁢

A

=

A

⁢

e

τ

⁢

A

=

e

τ

⁢

A

⁢

A

.

(2)

Consider another linear operator

B

. Let

B

⁢

(

τ

)

=

e

τ

⁢

A

⁢

B

⁢

e

-

τ

⁢

A

. Then one can prove the following series representation for

B

⁢

(

τ

)

:

B

⁢

(

τ

)

=

∑

m

=

0

∞

τ

m

m

!

⁢

B

m

,

(3)

where

B

m

=

[

A

,

B

]

m

:=

[

A

,

[

A

,

B

]

m

-

1

]

and

B

0

:=

B

.
A very important special case of eq. (

3

) is known as the

Baker-Campbell-Hausdorff (BCH)

formula. Namely, for

τ

=

1

we get:

e

A

⁢

B

⁢

e

-

A

=

∑

m

=

0

∞

1

m

!

⁢

B

m

.

(4)

Alternatively, this

expression

may be rewritten as

[

B

,

e

-

A

]

=

e

-

A

⁢

(

[

A

,

B

]

+

1

2

⁢

[

A

,

[

A

,

B

]

]

+

⋯

)

,

(5)

or

[

e

A

,

B

]

=

(

[

A

,

B

]

+

1

2

⁢

[

A

,

[

A

,

B

]

]

+

⋯

)

⁢

e

A

.

(6)

There is a descendent of the

BCH formula

, which often is also referred to as BCH
formula. It provides us with the

multiplication

law for two

exponentials

of linear operators: Suppose

[

A

,

[

A

,

B

]

]

=

[

B

,

[

B

,

A

]

]

=

0

. Then,

e

A

⁢

e

B

=

e

A

+

B

⁢

e

1

2

⁢

[

A

,

B

]

.

(7)

Thus, if we want to commute two exponentials, we get an extra factor

e

A

⁢

e

B

=

e

B

⁢

e

A

⁢

e

[

A

,

B

]

.

(8)

Title

Baker-Campbell-Hausdorff formula(e)

Canonical

name

BakerCampbellHausdorffFormulae

Date of creation

2013-03-22 13:39:51

Last modified on

2013-03-22 13:39:51

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

13

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 47A05

Synonym

BCH formula

Synonym

Baker-Campbell-Hausdorff formula

Synonym

Baker-Campbell-Hausdorff formulae

Generated on Fri Feb 9 21:44:52 2018 by

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bounded inverse theorem

bounded inverse theorem

The next result is a corollary of the

open mapping theorem

. It is often called the

bounded inverse theorem

or the

inverse mapping theorem

.

Theorem -

Let

X

,

Y

be

Banach spaces

. Let

T

:

X

⟶

Y

be an invertible

bounded operator

. Then

T

-

1

is also .

Proof :

T

is a surjective continuous operator between the Banach spaces

X

and

Y

. Therefore, by the open mapping theorem,

T

takes

open sets

to open sets.

So, for every open set

U

⊆

X

,

T

⁢

(

U

)

is open in

Y

.

Hence

(

T

-

1

)

-

1

⁢

(

U

)

is open in

Y

, which proves that

T

-

1

is

continuous

, i.e.

bounded

.

□

0.0.1

Remark

It is usually of great importance to know if a bounded operator

T

:

X

⟶

Y

has a bounded inverse. For example, suppose the equation

T

⁢

x

=

y

has unique solutions

x

for every given

y

∈

Y

. Suppose also that the above equation is very difficult to solve (numerically) for a given

y

0

, but easy to solve for a value

y

~

”near”

y

0

. Then, if

T

-

1

is continuous, the correspondent solutions

x

0

and

x

~

are also ”near” since

∥

x

0

-

x

~

∥

=

∥

T

-

1

⁢

y

0

-

T

-

1

⁢

y

~

∥

≤

∥

T

-

1

∥

⁢

∥

y

0

-

y

~

∥

Therefore we can solve the equation for a ”near” value

y

~

instead, without obtaining a significant error.

Title

bounded inverse theorem

Canonical name

BoundedInverseTheorem

Date of creation

2013-03-22 17:30:51

Last modified on

2013-03-22 17:30:51

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

11

Author

asteroid (17536)

Entry type

Corollary

Classification

msc 47A05

Classification

msc 46A30

Synonym

inverse mapping theorem

Generated on Fri Feb 9 21:44:53 2018 by

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closed operator

closed operator

Let

B

be a

Banach space

.
A

linear operator

A

:

𝒟

⁢

(

A

)

⊂

B

→

B

is said to be if
for every sequence

{

x

n

}

n

∈

ℕ

in

𝒟

⁢

(

A

)

converging to

x

∈

B

such that

A

⁢

x

n

→

n

→

∞

y

∈

B

, it holds

x

∈

𝒟

⁢

(

A

)

and

A

⁢

x

=

y

.
Equivalently,

A

is closed if its graph is closed in

B

⊕

B

.

Given an

operator

A

, not necessarily closed, if the

closure

of its graph in

B

⊕

B

happens to be the graph of some operator, we call that operator the

closure

of

A

, and we say that

A

is

closable

. We denote the closure of

A

by

A

¯

. It follows easily that

A

is the restriction of

A

¯

to

𝒟

⁢

(

A

)

.

A

core

of a

closable operator

is a subset

𝒞

of

𝒟

⁢

(

A

)

such that the closure of the restriction of

A

to

𝒞

is

A

¯

.

The following properties are easily checked:

1.

Any

bounded linear operator

defined on the whole space

B

is closed;

2.

If

A

is closed then

A

-

λ

⁢

I

is closed;

3.

If

A

is closed and it has an

inverse

, then

A

-

1

is also closed;

4.

An operator

A

admits a closure if and only if for every pair of sequences

{

x

n

}

and

{

y

n

}

in

𝒟

⁢

(

A

)

, both converging to

z

∈

B

, and such that both

{

A

⁢

x

n

}

and

{

A

⁢

y

n

}

converge, it holds

lim

n

⁡

A

⁢

x

n

=

lim

n

⁡

A

⁢

y

n

.

Title

closed operator

Canonical name

ClosedOperator

Date of creation

2013-03-22 13:48:20

Last modified on

2013-03-22 13:48:20

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

9

Author

Koro (127)

Entry type

Definition

Classification

msc 47A05

Synonym

closed

Defines

closure

Defines

closable

Defines

core

Generated on Fri Feb 9 21:44:56 2018 by

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densely defined

densely defined

Given a

topological space

X

, we say that a map

f

:

Y

→

X

is

densely defined

if its domain

Y

is a

dense subset

of

X

.

This terminology is commonly used in the of

linear operators

with the following meaning: In a

normed space

X

, a linear operator

A

:

𝒟

⁢

(

A

)

⊂

X

→

X

is said to be

densely defined

if

𝒟

⁢

(

A

)

is a dense

vector subspace

of

X

.

Title

densely defined

Canonical

name

DenselyDefined

Date of creation

2013-03-22 13:48:12

Last modified on

2013-03-22 13:48:12

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Definition

Classification

msc 47A05

Classification

msc 54C10

Generated on Fri Feb 9 21:44:57 2018 by

LaTeXML
existence of adjoints of bounded operators

existence of adjoints of bounded operators

Let

ℋ

be a

Hilbert space

and let

T

:

𝒟

⁢

(

T

)

⊂

ℋ

⟶

ℋ

be a

densely defined

linear operator

.

Theorem -

If

T

is

bounded

(

http://planetmath.org/ContinuousLinearMapping

) then its

adjoint

T

*

is everywhere defined and is also bounded.

Proof :

Since

T

is densely defined and bounded, it extends uniquely to a bounded (everywhere defined) linear operator on

ℋ

, which we denote by

T

~

.

For each

z

∈

ℋ

, the function

f

:

ℋ

⟶

ℂ

defined by

f

⁢

(

x

)

=

⟨

T

~

⁢

x

,

z

⟩

defines a bounded

linear functional

on

ℋ

. By the

Riesz representation theorem

there exists

u

∈

ℋ

such that

f

⁢

(

x

)

=

⟨

x

,

u

⟩

i.e.

⟨

T

~

⁢

x

,

z

⟩

=

⟨

x

,

u

⟩

.

Since

T

~

extends

T

, we also have that for every

z

∈

ℋ

there exists

u

∈

ℋ

such that

⟨

T

⁢

x

,

z

⟩

=

⟨

x

,

u

⟩

⁢

for every

⁢

x

∈

𝒟

⁢

(

T

)

.

We conclude that

T

*

is everywhere defined. To see that it is bounded one just needs to check that

sup

z

≠

0

⁡

∥

T

*

⁢

z

∥

∥

z

∥

=

sup

z

≠

0

T

*

⁢

z

≠

0

⁡

|

⟨

T

*

⁢

z

,

T

*

⁢

z

⟩

|

∥

T

*

⁢

z

∥

⁢

∥

z

∥

≤

sup

z

≠

0

x

≠

0

⁡

|

⟨

x

,

T

*

⁢

z

⟩

|

∥

x

∥

⁢

∥

z

∥

=

sup

z

≠

0

x

≠

0

⁡

|

⟨

T

⁢

x

,

z

⟩

|

∥

x

∥

⁢

∥

z

∥

≤

∥

T

∥

where the last inequality comes from the

Cauchy-Schwarz inequality

and the fact that

T

is bounded.

□

Remark -

This theorem shows in particular that

bounded linear operators

T

:

ℋ

⟶

ℋ

have bounded adjoints

T

*

:

ℋ

⟶

ℋ

.

Title

existence of adjoints of bounded operators

Canonical name

ExistenceOfAdjointsOfBoundedOperators

Date of creation

2013-03-22 17:33:44

Last modified on

2013-03-22 17:33:44

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 47A05

Synonym

bounded operators

have (bounded) adjoints

Generated on Fri Feb 9 21:45:00 2018 by

LaTeXML
polar decomposition

polar decomposition

The

polar decomposition

of an

operator

is a

generalization

of the familiar factorization of a

complex number

z

in a radial part

|

z

|

and an angular part

z

/

|

z

|

.

Let

ℋ

be a

Hilbert space

,

x

a

bounded operator

on

ℋ

. Then there exist a pair

(

h

,

u

)

, with

h

a

bounded

positive operator

and

u

a

partial isometry

on

ℋ

, such that

x

=

u

⁢

h

.

If we impose the further conditions that

1

-

u

*

⁢

u

is the projection to the kernel of

x

, and

ker

⁡

(

h

)

=

ker

⁡

(

x

)

, then

(

h

,

u

)

is unique, and is called the

polar decomposition

of

x

. The operator

h

will be

|

x

|

, the square root of

x

*

⁢

x

, and

u

will be the partial isometry, determined by

•

u

⁢

ξ

=

0

for

ξ

∈

ker

⁡

(

x

)

•

u

⁢

(

|

x

|

⁢

ξ

)

=

x

⁢

ξ

for

ξ

∈

ℋ

.

If

x

is a closed,

densely defined

unbounded operator on

ℋ

, the polar decomposition

(

u

,

h

)

still exists, where now

h

will be the

unbounded

positive operator

|

x

|

with the same domain

𝒟

⁢

(

x

)

as

x

, and

u

still the partial isometry determined by

•

u

⁢

ξ

=

0

for

ξ

∈

ker

⁡

(

x

)

•

u

⁢

(

|

x

|

⁢

ξ

)

=

x

⁢

ξ

for

ξ

∈

𝒟

⁢

(

x

)

.

If

x

is affiliated with a

von Neumann algebra

M

, both

u

and

h

will be affiliated with

M

.

Title

polar decomposition

Canonical name

PolarDecomposition

Date of creation

2013-03-22 16:01:54

Last modified on

2013-03-22 16:01:54

Owner

aube (13953)

Last modified by

aube (13953)

Numerical id

10

Author

aube (13953)

Entry type

Definition

Classification

msc 47A05

Generated on Fri Feb 9 21:45:02 2018 by

LaTeXML
polar decomposition in von Neumann algebras

polar decomposition in von Neumann algebras

-

Let

ℳ

be a

von Neumann algebra

acting on a

Hilbert space

H

and

T

∈

ℳ

. If

T

=

V

⁢

R

is the

polar decomposition

for

T

with

K

⁢

e

⁢

r

⁢

V

=

K

⁢

e

⁢

r

⁢

R

, then both

V

and

R

belong to

ℳ

.

Proof :

•

As

ℳ

is a

C

*

-

algebra

(

http://planetmath.org/CAlgebra

), it is known that

R

=

T

*

⁢

T

belongs to

ℳ

. (proof will be added later)

•

To see that

V

also belongs to

ℳ

, by the

double commutant theorem

, it suffices to show that

V

belongs to

ℳ

′′

(the

double commutant

of

ℳ

).

Suppose

S

∈

ℳ

′

. We intend to prove that

V

commutes with

S

.

For

x

∈

H

we have that

T

⁢

S

⁢

x

=

S

⁢

T

⁢

x

=

S

⁢

V

⁢

R

⁢

x

and

T

⁢

S

⁢

x

=

V

⁢

R

⁢

S

⁢

x

=

V

⁢

S

⁢

R

⁢

x

So

S

⁢

V

and

V

⁢

S

agree on

R

⁢

a

⁢

n

⁢

R

¯

.

As

R

is

self-adjoint

,

R

⁢

a

⁢

n

⁢

R

¯

⟂

=

K

⁢

e

⁢

r

⁢

R

, and so it remains to show that

S

⁢

V

and

V

⁢

S

agree on

K

⁢

e

⁢

r

⁢

R

. Recall that, by

hypothesis

,

K

⁢

e

⁢

r

⁢

R

=

K

⁢

e

⁢

r

⁢

V

.

Let

x

∈

K

⁢

e

⁢

r

⁢

R

. We have that

R

⁢

S

⁢

x

=

S

⁢

R

⁢

x

=

0

and therefore

S

⁢

(

K

⁢

e

⁢

r

⁢

R

)

⊆

K

⁢

e

⁢

r

⁢

R

=

K

⁢

e

⁢

r

⁢

V

and so we can conclude that

V

⁢

S

is identically zero in

K

⁢

e

⁢

r

⁢

R

.

Clearly

S

⁢

V

is also identically zero on

K

⁢

e

⁢

r

⁢

R

=

K

⁢

e

⁢

r

⁢

V

.

Thus

V

⁢

S

and

S

⁢

V

agree on

K

⁢

e

⁢

r

⁢

R

. Therefore

S

⁢

V

=

V

⁢

S

and so

V

∈

ℳ

′′

=

ℳ

⁢

□

Title

polar decomposition in von Neumann algebras

Canonical

name

PolarDecompositionInVonNeumannAlgebras

Date of creation

2013-03-22 17:28:54

Last modified on

2013-03-22 17:28:54

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Result

Classification

msc 47A05

Classification

msc 46L10

Generated on Fri Feb 9 21:45:04 2018 by

LaTeXML
properties of the adjoint operator

properties of the adjoint operator

Let

A

and

B

be

linear operators

in a

Hilbert space

,
and let

λ

∈

ℂ

. Assuming all the

operators

involved are

densely defined

, the following properties hold:

1.

If

A

-

1

exists and is densely defined, then

(

A

-

1

)

*

=

(

A

*

)

-

1

;

2.

(

λ

⁢

A

)

*

=

λ

¯

⁢

A

*

;

3.

A

⊂

B

implies

B

*

⊂

A

*

;

4.

A

*

+

B

*

⊂

(

A

+

B

)

*

;

5.

B

*

⁢

A

*

⊂

(

A

⁢

B

)

*

;

6.

(

A

+

λ

⁢

I

)

*

=

A

*

+

λ

¯

⁢

I

;

7.

A

*

is a

closed operator

.

Remark.

The notation

A

⊂

B

for operators means that

B

is an of

A

, i.e.

A

is the
restriction (

http://planetmath.org/RestrictionOfAFunction

) of

B

to a smaller domain.

Also, we have the following

Proposition 1

If

A

admits a

closure

(

http://planetmath.org/ClosedOperator

)

A

¯

, then

A

*

is densely defined and

(

A

*

)

*

=

A

¯

.

Title

properties of the

adjoint operator

Canonical name

PropertiesOfTheAdjointOperator

Date of creation

2013-03-22 13:48:14

Last modified on

2013-03-22 13:48:14

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

12

Author

Koro (127)

Entry type

Theorem

Classification

msc 47A05

Generated on Fri Feb 9 21:45:06 2018 by

LaTeXML
transpose operator

transpose operator

Let

X

,

Y

be

normed vector spaces

and

X

′

,

Y

′

be their

continuous dual

spaces.

-

Let

T

:

X

⟶

Y

be a

bounded linear operator

. The

operator

T

′

:

Y

′

⟶

X

′

given by

T

′

⁢

ϕ

=

ϕ

∘

T

,

ϕ

∈

Y

′

is called the

transpose operator

of

T

or the

conjugate operator

of

T

.

It is clear that

T

′

is well defined, i.e.

ϕ

∘

T

∈

X

′

, since the composition of two continuous linear operators is again a continuous linear operator.

Moreover, it can be easily checked that

T

′

is a bounded linear operator.

Remarks -

•

When the

vector spaces

are finite dimensional, the transpose operator corresponds to transposing (

http://planetmath.org/

Transpose

) the matrix associated to it.

•

For

Hilbert spaces

, a somewhat

similar

definition

is that of

adjoint operator

. But this two notions do not coincide: while the transpose operator corresponds to the transpose of a matrix, the adjoint operator corresponds to the

conjugate transpose

of a matrix.

Title

transpose operator

Canonical

name

TransposeOperator

Date of creation

2013-03-22 17:34:19

Last modified on

2013-03-22 17:34:19

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Definition

Classification

msc 47A05

Classification

msc 46-00

Synonym

conjugate operator

Related topic

Transpose

Related topic

Adjoint5

Generated on Fri Feb 9 21:45:08 2018 by

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