aarjaneiro/planetmath huggingface dataset:
triple scalar product

triple scalar product

The triple scalar product of three vectors is defined in

ℝ

3

as

det

⁡

[

a

1

b

1

c

1

a

2

b

2

c

2

a

3

b

3

c

3

]

=

a

→

⋅

(

b

→

×

c

→

)

=

a

1

⁢

det

⁡

[

b

2

c

2

b

3

c

3

]

-

a

2

⁢

det

⁡

[

b

1

c

1

b

3

c

3

]

+

a

3

⁢

det

⁡

[

b

1

c

1

b

2

c

2

]

The

determinant

above is

positive

if the three vectors

satisfy

the

right-hand rule

and negative otherwise. Recall that the magnitude of the

cross product

of two vectors is

equivalent

to the area of the

parallelogram

they form, and the

dot product

is equivalent to the

product

of the

projection

of one vector onto another with the length of the vector projected upon. Putting these two ideas together, we can see that

|

a

→

⋅

(

b

→

×

c

→

)

|

=

|

b

→

×

c

→

|

⁢

|

a

→

|

⁢

cos

⁡

θ

=

base

⋅

height

=

Volume of parallelepiped

Thus, the magnitude of the triple scalar product is equivalent to the volume of the

parallelepiped

formed by the three vectors. It follows that the triple scalar product of three

coplanar

or collinear vectors is then 0. When we keep the sign of the volume, we

preserve

orientation.

\PMlinktofile

View SVGTripleScalarProduct.svg

Note that there is no

rigid motion

, barring

reflection

, that will allow you to move/rotate the left parallelepiped so that the respective vectors coincide. This is the sense of orientation that is used in

differential geometry

.

Identities

related to the triple scalar product:

•

(

A

→

×

B

→

)

⋅

C

→

=

(

B

→

×

C

→

)

⋅

A

→

=

(

C

→

×

A

→

)

⋅

B

→

•

A

→

⋅

(

B

→

×

C

→

)

=

-

A

→

⋅

(

C

→

×

B

→

)

The latter is implied by the

properties

of the cross product.

Generalization

:

Given a metrizable

manifold

M

, the scalar n-product of an ordered list of n vectors

(

v

1

→

,

v

2

→

,

…

,

v

n

→

)

corresponds to the signed n-volume of the n-paralleletope spanned by the vectors and is calculated by:

∗

(

⋁

i

=

1

n

v

i

→

)

where

∗

is the

Hodge star operator

and

∨

is the

exterior product

. Note that this simply corresponds to the determinant of the list in n-dimensional

Euclidean space

.

Title

triple scalar product

Canonical

name

TripleScalarProduct

Date of creation

2013-03-22 11:59:24

Last modified on

2013-03-22 11:59:24

Owner

slider142 (78)

Last modified by

slider142 (78)

Numerical id

15

Author

slider142 (78)

Entry type

Definition

Classification

msc 47A05

Classification

msc 46M05

Classification

msc 11H46

Synonym

scalar triple product

Synonym

determinant in R^3

Related topic

DotProduct

Related topic

CrossProduct

Generated on Fri Feb 9 21:45:10 2018 by

LaTeXML
canonical basis for symmetric bilinear forms

canonical basis for symmetric bilinear forms

If

B

:

V

×

V

→

K

is a

symmetric bilinear form

over a

finite-dimensional

vector space

, where the characteristic of the field is
not 2,
then we may prove that there is an

orthogonal basis

such that

B

is represented by

a

1

0

…

0

0

a

2

…

0

⋮

⋮

⋱

⋮

0

0

…

a

n

Recall that a

bilinear form

has a well-defined rank, and denote this by

r

.

If

K

=

ℝ

we may choose a basis such that

a

1

=

⋯

=

a

t

=

1

,

a

t

+

1

=

⋯

=

a

t

+

p

=

-

1

and

a

t

+

p

+

j

=

0

, for some

integers

p

and

t

,
where

1

≤

j

≤

n

-

t

-

p

.
Furthermore, these integers are

invariants

of the bilinear form.
This is known as

Sylvester’s Law of Inertia

.

B

is

positive definite

if and only if

t

=

n

,

p

=

0

. Such a form constitutes a

real

inner product space

.

If

K

=

ℂ

we may go further and choose a basis such that

a

1

=

⋯

=

a

r

=

1

and

a

r

+

j

=

0

, where

1

≤

j

≤

n

-

r

.

If

K

=

F

p

we may choose a basis such that

a

1

=

⋯

=

a

r

-

1

=

1

,

a

r

=

n

or

a

r

=

1

;
and

a

r

+

j

=

0

, where

1

≤

j

≤

n

-

r

, and

n

is the least

positive

quadratic non-residue

.

Title

canonical basis for symmetric bilinear forms

Canonical name

CanonicalBasisForSymmetricBilinearForms

Date of creation

2013-03-22 14:56:25

Last modified on

2013-03-22 14:56:25

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 47A07

Classification

msc 11E39

Classification

msc 15A63

Defines

Sylvester’s Law of Inertia

Generated on Fri Feb 9 21:45:17 2018 by

LaTeXML
Hermitian form

Hermitian form

A

sesquilinear form

over a pair of

complex vector spaces

(

V

,

W

)

is a

function

B

:

V

×

W

→

ℂ

satisfying the following properties:

1.

B

⁢

(

𝐯

1

+

𝐯

2

,

𝐰

)

=

B

⁢

(

𝐯

1

,

𝐰

)

+

B

⁢

(

𝐯

2

,

𝐰

)

2.

B

⁢

(

𝐯

,

𝐰

1

+

𝐰

2

)

=

B

⁢

(

𝐯

,

𝐰

1

)

+

B

⁢

(

𝐯

,

𝐰

2

)

3.

B

⁢

(

c

⁢

𝐯

,

d

⁢

𝐰

)

=

c

⁢

B

⁢

(

𝐯

,

𝐰

)

⁢

d

¯

for all

𝐯

,

𝐯

1

,

𝐯

2

∈

V

,

𝐰

,

𝐰

1

,

𝐰

2

∈

W

, and

c

,

d

∈

ℂ

. The

vector spaces

V

and

W

are often identical, although the definition does not require them to be the same vector space.

A sesquilinear form

B

:

V

×

V

→

ℂ

over a single vector space

V

is called a

Hermitian form

if it is

complex conjugate

symmetric

: namely, if

B

⁢

(

𝐯

1

,

𝐯

2

)

=

B

⁢

(

𝐯

2

,

𝐯

1

)

¯

.

An

inner product

over a complex vector space is a

positive definite

Hermitian form.

Title

Hermitian form

Canonical name

HermitianForm

Date of creation

2013-03-22 12:25:47

Last modified on

2013-03-22 12:25:47

Owner

djao (24)

Last modified by

djao (24)

Numerical id

8

Author

djao (24)

Entry type

Definition

Classification

msc 47A07

Classification

msc 15A63

Classification

msc 11E39

Synonym

sesquilinear form

Synonym

sesqui-linear form

Related topic

InnerProduct

Generated on Fri Feb 9 21:45:19 2018 by

LaTeXML
non-degenerate bilinear form

non-degenerate bilinear form

A

bilinear form

B

over a

vector space

V

is said to be

non-degenerate

when

•

if

B

⁢

(

𝐱

,

𝐲

)

=

0

for all

𝐱

∈

V

, then

𝐲

=

𝟎

, and

•

if

B

⁢

(

𝐱

,

𝐲

)

=

0

for all

𝐲

∈

V

, then

𝐱

=

𝟎

.

Title

non-degenerate bilinear form

Canonical

name

NondegenerateBilinearForm

Date of creation

2013-03-22 12:25:56

Last modified on

2013-03-22 12:25:56

Owner

djao (24)

Last modified by

djao (24)

Numerical id

5

Author

djao (24)

Entry type

Definition

Classification

msc 47A07

Classification

msc 15A63

Classification

msc 11E39

Synonym

non-degenerate form

Synonym

nondegenerate bilinear form

Synonym

nondegenerate form

Synonym

non-degenerate

Synonym

nondegenerate

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

LaTeXML
Riesz representation theorem of bounded sesquilinear forms

Riesz representation theorem of bounded sesquilinear forms

Bounded sesquilinear forms

Let

H

1

,

H

2

be two

Hilbert spaces

.

Definition -

A

sesquilinear form

[

⋅

,

⋅

]

:

H

1

×

H

2

→

ℂ

is said to be

bounded

if there is a constant

C

≥

0

such that

[

ξ

,

η

]

≤

C

⁢

∥

ξ

∥

⁢

∥

η

∥

for all

ξ

∈

H

1

and

η

∈

H

2

.

Bounded sesquilinear forms

are precisely those which are

continuous

from

H

1

×

H

2

to

ℂ

.

Examples :

•

When

H

1

and

H

2

are the same Hilbert space, denoted by

H

, the

inner product

⟨

⋅

,

⋅

⟩

in

H

is itself a bounded sesquilinear form. The boundedness condition follows from the

Cauchy-Schwarz inequality

.

•

Let

T

:

H

1

→

H

2

be a

bounded linear operator

and denote by

⟨

⋅

,

⋅

⟩

the inner product in

H

2

. The function

[

⋅

,

⋅

]

:

H

1

×

H

2

→

ℂ

defined by

[

ξ

,

η

]

:=

⟨

T

⁢

ξ

,

η

⟩

is a bounded sesquilinear form. The boundedness condition follows from the Cauchy-Schwarz inequality and the fact that

T

is bounded.

Riesz representation of bounded sesquilinear forms

The second example above is in fact the general case. To every bounded sesquilinear form one can

associate

to it a unique

bounded operator

. That is content of the following result:

Theorem - Riesz -

Let

H

1

,

H

2

be two Hilbert spaces and denote by

⟨

⋅

,

⋅

⟩

the inner product in

H

2

. For every bounded sesquilinear form

[

⋅

,

⋅

]

:

H

1

×

H

2

→

ℂ

there is a unique bounded linear operator

T

:

H

1

→

H

2

such that

[

ξ

,

η

]

=

⟨

T

⁢

ξ

,

η

⟩

,

ξ

∈

H

1

,

η

∈

H

2

.

Thus, there is a correspondence between bounded linear operators and bounded sesquilinear forms. Actually, in the early twentieth century,

spectral theory

was formulated solely in terms of sesquilinear forms on Hilbert spaces. Only later it was realized that this could be achieved, perhaps in a more intuitive manner, by considering

linear operators

instead. The linear operator approach has its advantages, as for example one can define the composition of linear operators but not of sesquilinear forms. Nevertheless it is many times useful to define a linear operator by specifying its sesquilinear form.

Title

Riesz representation theorem of bounded sesquilinear forms

Canonical name

RieszRepresentationTheoremOfBoundedSesquilinearForms

Date of creation

2013-03-22 18:41:38

Last modified on

2013-03-22 18:41:38

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

10

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 47A07

Classification

msc 46C05

Synonym

Riesz lemma on bounded sesquilinear forms

Synonym

correspondence between bounded operators and bounded sesquilinear forms

Defines

bounded sesquilinear form

Generated on Fri Feb 9 21:45:23 2018 by

LaTeXML
sesquilinear forms over general fields

sesquilinear forms over general fields

Let

V

be a

vector space

over a field

k

.

k

may be of any

characteristic

.

1

Sesquilinear Forms

Definition 1

.

A function

b

:

V

×

V

→

k

is sesquilinear if it satisfies each of
the following:

1.

b

⁢

(

v

,

w

+

u

)

=

b

⁢

(

v

,

w

)

+

b

⁢

(

v

,

u

)

and

b

⁢

(

v

+

u

,

w

)

=

b

⁢

(

v

,

w

)

+

b

⁢

(

u

,

w

)

for all

u

,

v

,

w

∈

V

;

2.

For a given

field automorphism

θ

of

k

,

b

⁢

(

v

,

l

⁢

w

)

=

l

θ

⁢

b

⁢

(

v

,

w

)

and

b

⁢

(

l

⁢

v

,

w

)

=

l

⁢

b

⁢

(

v

,

w

)

for all

v

,

w

∈

V

and

l

∈

k

.

Remark 2

.

It is possible to apply the field automorphism in the first

variable

but is more common to do so in the second variable. Also, if

θ

=

1

the form is a

bilinear form

.

Sesquilinear forms

are commonly ascribed any

combination

of the following

properties

:

•

non-degenerate

,

•

reflexive

, (commonly required to define

perpendicular

);

•

positive definite

(this condition requires the

fixed field

of

θ

,

k

0

, be an

ordered field

, such as the

rationals

ℚ

or reals

ℝ

).

Non-degenerate sesquilinear and bilinear forms apply to

projective geometries

as

dualities

and

polarities

through the induced

⟂

operation. (See polarity (

http://planetmath.org/Polarity2

).)

2

Hermitian Forms

If

θ

2

=

1

, it is common to exchange notation at this point and use the same notation of

l

¯

for

l

θ

as is common for

complex conjugation

– even if

k

is not

ℂ

. Then

l

¯

¯

=

l

.

In this notation,

Hermitian forms

may be defined by the property

b

⁢

(

v

,

w

)

=

b

⁢

(

w

,

v

)

¯

.

Remark 3

.

It is not uncommon to see

hermitian

or

Hermitean

instead of Hermitian. The name is a tribute to Charles Hermite of the Ecole Polytechnique.

Title

sesquilinear forms over general fields

Canonical

name

SesquilinearFormsOverGeneralFields

Date of creation

2013-03-22 15:58:17

Last modified on

2013-03-22 15:58:17

Owner

Algeboy (12884)

Last modified by

Algeboy (12884)

Numerical id

11

Author

Algeboy (12884)

Entry type

Definition

Classification

msc 47A07

Classification

msc 15A63

Classification

msc 11E39

Classification

msc 51A05

Synonym

Hermitian form

Synonym

Hermitean form

Related topic

ReflexiveNonDegenerateSesquilinear

Related topic

NonDegenerate

Related topic

Polarity2

Related topic

Projectivity

Related topic

ProjectiveGeometry

Related topic

Isometry2

Related topic

ProjectiveGeometry3

Related topic

ClassicalGroups

Defines

sesquilinear form

Defines

Hermitian form

Defines

bilinear form

Defines

Hermitean

Generated on Fri Feb 9 21:45:25 2018 by

LaTeXML
resolvent matrix

resolvent matrix

The

resolvent matrix

of a matrix

A

is defined as

R

A

⁢

(

s

)

=

(

s

⁢

I

-

A

)

-

1

.

Note:

I

is the

identity matrix

and

s

is a

complex variable

. Also note that

R

A

⁢

(

s

)

is undefined on

S

⁢

p

⁢

(

A

)

(the

spectrum

of

A

).

More generally, let

A

be a unital algebra over the field of

complex numbers

ℂ

. The

resolvent

R

x

of an element

x

∈

A

is a function from

ℂ

-

S

⁢

p

⁢

(

x

)

to

A

given by

R

x

⁢

(

s

)

=

(

s

⋅

1

-

x

)

-

1

where

S

⁢

p

⁢

(

x

)

is the spectrum of

x

:

S

⁢

p

⁢

(

x

)

=

{

t

∈

ℂ

∣

t

⋅

1

-

x

⁢

is not invertible in

⁢

A

}

.

If

A

is

commutative

and

s

∉

S

⁢

p

⁢

(

x

)

∪

S

⁢

p

⁢

(

y

)

, then

R

x

⁢

(

s

)

-

R

y

⁢

(

s

)

=

R

x

⁢

(

s

)

⁢

R

y

⁢

(

s

)

⁢

(

x

-

y

)

.

Title

resolvent matrix

Canonical name

ResolventMatrix

Date of creation

2013-03-22 13:36:20

Last modified on

2013-03-22 13:36:20

Owner

mps (409)

Last modified by

mps (409)

Numerical id

8

Author

mps (409)

Entry type

Definition

Classification

msc 47A10

Classification

msc 15A15

Defines

resolvent

Generated on Fri Feb 9 21:45:27 2018 by

LaTeXML
Dunkl-Williams inequality

Dunkl-Williams inequality

Let

V

be an

inner product space

and

a

,

b

∈

V

. If

a

≠

0

and

b

≠

0

, then

∥

a

-

b

∥

≥

1

2

⁢

(

∥

a

∥

+

∥

b

∥

)

⁢

∥

a

∥

a

∥

-

b

∥

b

∥

∥

.

(1)

Equality holds if and only if

a

=

0

,

b

=

0

,

∥

a

∥

=

∥

b

∥

or

a

⁢

∥

b

∥

=

b

⁢

∥

a

∥

.
In fact, if (1) holds
and

V

is a

normed linear space

, then

V

is an inner product space.

If

X

is a normed linear space and

a

≠

0

and

b

≠

0

then

∥

a

-

b

∥

≥

1

4

⁢

(

∥

a

∥

+

∥

b

∥

)

⁢

∥

a

∥

a

∥

-

b

∥

b

∥

∥

.

(2)

Equality holds if and only if

a

=

0

,

b

=

0

or

a

=

b

.
The constant

1

4

is best possible. For example, let

X

be the set of ordered pairs of

real numbers

,
with norm of

(

x

1

,

x

2

)

equal to

|

x

1

|

+

|

x

2

|

.

Let

a

=

(

1

,

ϵ

)

and

b

=

(

1

,

0

)

where

ϵ

is a small positive number. After a bit
of routine calculation, it is easily seen that the best possible constant is

1

4

.

The

inequality

(2) has been generalized in the case where

X

is a normed linear space
over the reals. In that case one can show:

∥

a

-

b

∥

≥

c

p

⁢

(

∥

a

∥

p

+

∥

b

∥

p

)

1

/

p

⁢

∥

a

∥

a

∥

-

b

∥

b

∥

∥

(3)

where

c

p

=

2

-

1

-

1

/

p

if

0

<

p

≤

1

and

c

p

=

1

/

4

if

p

≥

1

. The case

p

=

1

is the
Dunkl and Williams inequality.

If

X

is a normed linear space and

0

<

p

≤

1

then (3) holds with

c

p

=

2

-

1

/

p

if and only if

X

is an inner product space.

The inequality (2) can be improved slightly to get:

∥

a

-

b

∥

≥

1

2

⁢

max

⁡

(

∥

a

∥

,

∥

b

∥

)

⁢

∥

a

∥

a

∥

-

b

∥

b

∥

∥

.

(4)

Equality holds in (4) if and only if

a

and

b

span an

ℓ

2

1

in the underlying real vector space with

±

∥

b

-

a

∥

-

1

⁢

(

b

-

a

)

and

±

∥

a

∥

-

1

⁢

a

(or

±

∥

b

∥

-

1

⁢

b

) as the vertices of the unit
parallelogram.

References

1

C.F. Dunkl, K.S. Williams,

A simple norm inequality

. Amer. Math. Monthly,

71

(1) (1964) 53-54.

2

W.A. Kirk, M.F. Smiley,

Another

characterization

of inner product spaces

, Amer. Math. Monthly,

71

, (1964), 890-891.

3

A.M. Alrashed,

Norm inequalities and Characterizations of Inner Product Spaces

,
J. Math. Anal. Appl.

176

, (2) (1993), 587-593.

4

J.L. Massera, J.J. Schäffer,

Linear differential equations and

functional analysis

, Annals of Math.

67

(2)(1958), 517-573. (on page 538)

5

L.M. Kelly,

The Massera-Schäeffer equality

, Amer. Math. Monthly,

73

, (1966) 1102-1103.

Title

Dunkl-Williams inequality

Canonical

name

DunklWilliamsInequality

Date of creation

2013-03-22 16:56:38

Last modified on

2013-03-22 16:56:38

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

12

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 47A12

Generated on Fri Feb 9 21:45:30 2018 by

LaTeXML
invariant subspace problem

invariant subspace problem

Initially formulated for

Banach spaces

, the

invariant subspace conjecture

stated the following:

Let

X

be a complex Banach space. Then every

bounded operator

T

in

X

has a non-trivial closed (

http://planetmath.org/ClosedSet

)

invariant subspace

, i.e. there exists a closed

vector subspace

S

⊂

X

such that

S

≠

0

,

S

≠

X

and

T

⁢

(

S

)

⊆

S

.

This

conjecture

was proven to be false when P. Enflo (1975) and . Read (1984) gave examples of bounded operators which did not have the above

property

.

However, if one considers only

Hilbert spaces

, this is still an open problem. Today the

invariant subspace conjecture

is formulated as follows:

Let

H

be a complex Hilbert space. Then every bounded operator

T

in

H

has a non-trivial invariant subspace, i.e. there exists a closed vector subspace

S

⊂

H

such that

S

≠

0

,

S

≠

H

and

T

⁢

(

S

)

⊆

S

.

Title

invariant subspace problem

Canonical

name

InvariantSubspaceProblem

Date of creation

2013-03-22 17:24:02

Last modified on

2013-03-22 17:24:02

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Conjecture

Classification

msc 47A15

Classification

msc 46-00

Synonym

invariant subspace conjecture

Generated on Fri Feb 9 21:45:31 2018 by

LaTeXML
ergodicity of a map in terms of its induced operator

ergodicity of a map in terms of its induced operator

Theorem

-

Let

(

X

,

𝔅

,

μ

)

be a

probability space

and

T

:

X

⟶

X

a

measure-preserving transformation

. The following statements are

equivalent

:

1.

-

T

is

ergodic

.

2.

- If

f

is a

measurable function

and

f

∘

T

=

f

a.e. (

http://planetmath.org/AlmostSurely

), then

f

is

constant

a.e.

3.

- If

f

is a measurable function and

f

∘

T

≥

f

a.e., then

f

is constant a.e.

4.

- If

f

∈

L

2

⁢

(

X

)

and

f

∘

T

=

f

a.e., then

f

is constant a.e..

5.

- If

f

∈

L

p

⁢

(

X

)

, with

p

≥

1

, and

f

∘

T

=

f

a.e., then

f

is constant a.e.

Let

U

T

denote the

operator

induced by

T

(

http://planetmath.org/OperatorInducedByAMeasurePreservingMap

), i.e. the operator defined by

U

T

⁢

f

:=

f

∘

T

. The statements above are statements about

U

T

. The above theorem can be rewritten as follows:

Theorem -

Let

(

X

,

𝔅

,

μ

)

be a probability space and

T

:

X

⟶

X

a measure-preserving transformation. The following statements are equivalent:

1.

-

T

is ergodic.

2.

- The only

fixed points

of

U

T

are the

functions

that are constant a.e.

3.

- If

f

a measurable function and

U

T

⁢

f

≥

f

a.e., then

f

is constant a.e.

4.

- The eigenspace of

U

T

(seen as an operator in

L

p

⁢

(

X

)

, with

p

≥

1

) associated with the eigenvalue

1

, is one-dimensional and consists of functions that are constant a.e.

Title

ergodicity of a map in terms of its induced operator

Canonical name

ErgodicityOfAMapInTermsOfItsInducedOperator

Date of creation

2013-03-22 17:59:22

Last modified on

2013-03-22 17:59:22

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 47A35

Classification

msc 37A30

Classification

msc 37A25

Classification

msc 28D05

Generated on Fri Feb 9 21:45:33 2018 by

LaTeXML
operator induced by a measure preserving map

operator induced by a measure preserving map

\PMlinkescapephrase

theorem

\PMlinkescapephrase

induced

\PMlinkescapephrase

property

\PMlinkescapephrase

properties

\PMlinkescapephrase

side

1

Induced Operators

Let

(

X

1

,

𝔅

1

,

μ

1

)

and

(

X

2

,

𝔅

2

,

μ

2

)

be

measure spaces

and denote by

L

0

⁢

(

X

1

)

and

L

0

⁢

(

X

2

)

the corresponding spaces of

measurable functions

(with values in

ℂ

).

Definition -

If

T

:

X

1

⟶

X

2

is a

measure-preserving transformation

we can define an

operator

U

T

:

L

0

⁢

(

X

2

)

⟶

L

0

⁢

(

X

1

)

by

(

U

T

⁢

f

)

⁢

(

x

)

:=

f

⁢

(

T

⁢

x

)

,

f

∈

L

0

⁢

(

X

2

)

,

x

∈

X

1

The operator

U

T

is called the

by

T

.

Many ideas in

ergodic theory

can be explored by studying this operator.

2

Basic Properties

The following are clear:

•

U

T

is linear.

•

U

T

maps real valued

functions

to real valued functions.

•

If

f

≥

0

then

U

T

⁢

f

≥

0

•

U

T

⁢

k

=

k

for every

constant function

k

.

•

U

T

⁢

(

f

⁢

g

)

=

U

T

⁢

(

f

)

⁢

U

T

⁢

(

g

)

.

•

U

T

maps

characteristic functions

to characteristic functions. Moreover,

U

T

⁢

χ

B

=

χ

T

-

1

⁢

B

, for every

measurable set

B

∈

𝔅

2

.

•

If

T

1

:

X

1

⟶

X

2

and

T

2

:

X

2

⟶

X

3

are measure preserving maps, then

U

T

2

∘

T

1

=

U

T

1

⁢

U

T

2

.

3

Preserving Integrals

Theorem 1 -

If

f

∈

L

0

⁢

(

X

2

)

then

∫

X

1

U

T

⁢

f

⁢

𝑑

μ

1

=

∫

X

2

f

⁢

𝑑

μ

2

, where if one side does not exist or is

infinite

, then the other side has the same property.

4

Induced Isometries

It can further be seen that a measure-preserving transformation induces an isometry between

L

p

-spaces (

http://planetmath.org/LpSpace

), for

p

≥

1

.

Theorem 2 -

Let

p

≥

1

. We have that

U

T

⁢

(

L

p

⁢

(

X

2

)

)

⊆

L

p

⁢

(

X

1

)

and, moreover,

∥

U

T

⁢

(

f

)

∥

p

=

∥

f

∥

p

,

for all

⁢

f

∈

L

p

⁢

(

X

2

)

Thus, when restricted to

L

p

-spaces,

U

T

is called the

isometry induced by

T

.

Title

operator induced by a measure preserving map

Canonical name

OperatorInducedByAMeasurePreservingMap

Date of creation

2013-03-22 17:59:19

Last modified on

2013-03-22 17:59:19

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Definition

Classification

msc 47A35

Classification

msc 28D05

Classification

msc 37A05

Defines

isometry induced by a measure preserving map

Generated on Fri Feb 9 21:45:36 2018 by

LaTeXML
Fredholm index

Fredholm index

Let

P

be a

Fredholm operator

.
The

index

of

P

is defined as

index

(

P

)

=

dim

⁡

ker

⁡

(

P

)

-

dim

⁡

coker

(

P

)

=

dim

⁡

ker

⁡

(

P

)

-

dim

⁡

ker

⁡

(

P

*

)

.

Note: this is well defined as

ker

⁡

(

P

)

and

ker

⁡

(

P

*

)

are

finite-dimensional

vector spaces

, for

P

Fredholm.

•

index

(

P

*

)

=

-

index

(

P

)

.

•

index

(

P

+

K

)

=

index

(

P

)

for any

compact operator

K

.

•

If

P

1

:

ℋ

1

→

ℋ

2

and

P

2

:

ℋ

2

→

ℋ

3

are Fredholm operators, then

index

(

P

2

⁢

P

1

)

=

index

(

P

1

)

+

index

(

P

2

)

.

•

If

t

→

P

t

,

t

∈

[

0

,

1

]

is a norm

continuous

path of Fredholm operators,
then

index

(

P

t

)

=

index

(

P

0

)

.

Fredholm operators of the form

𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑏𝑙𝑒

+

𝑐𝑜𝑚𝑝𝑎𝑐𝑡

have index zero.

Title

Fredholm index

Canonical

name

FredholmIndex

Date of creation

2013-03-22 13:20:45

Last modified on

2013-03-22 13:20:45

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

9

Author

mhale (572)

Entry type

Definition

Classification

msc 47A53

Synonym

index

Related topic

FredholmOperator

Generated on Fri Feb 9 21:45:38 2018 by

LaTeXML