aarjaneiro/planetmath huggingface dataset:
Fredholm operator

Fredholm operator

A

Fredholm operator

is a

bounded operator

between

Banach spaces

that has a finite dimensional kernel and cokernel (and closed range).
Equivalently, it is

invertible

modulo

compact operators

.
That is, if

F

:

X

→

Y

is a Fredholm operator between two

vector spaces

X

and

Y

,
then there exists a bounded operator

G

:

Y

→

X

such that

G

⁢

F

-

1

⁢

I

X

∈

𝕂

⁢

(

X

)

,

F

⁢

G

-

1

⁢

I

Y

∈

𝕂

⁢

(

Y

)

,

(1)

where

𝕂

⁢

(

X

)

denotes the space of compact operators on

X

.
(Another way to say this is that

F

is invertible in the Calkin algebra).
The set of Fredholm operators

{

F

:

X

→

X

}

is an open subset of the

Banach algebra

of bounded operators

{

T

:

X

→

X

}

.

If

F

is Fredholm then so is its

adjoint

,

F

*

.
If

T

∈

𝕂

⁢

(

X

,

Y

)

is a compact operator then

F

+

T

is also Fredholm.

Title

Fredholm operator

Canonical name

FredholmOperator

Date of creation

2013-03-22 12:58:52

Last modified on

2013-03-22 12:58:52

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

15

Author

mhale (572)

Entry type

Definition

Classification

msc 47A53

Related topic

FredholmIndex

Generated on Fri Feb 9 21:45:40 2018 by

LaTeXML
semi-Fredholm operator

semi-Fredholm operator

A

semi-Fredholm operator

is a

bounded operator

between

Banach spaces

that has a finite dimensional kernel or cokernel, and closed range.
There are two semigroups of semi-Fredholm operators:
upper semi-Fredholm operators (those which have finite dimensional kernels),
and lower semi-Fredholm operators (those which have finite dimensional cokernels).
A semi-Fredholm operator that is both upper and lower is Fredholm.
If an

operator

is homotopic through (a norm

continuous

path of) semi-Fredholm operators to a

Fredholm operator

, then it is Fredholm.

Title

semi-Fredholm operator

Canonical name

SemiFredholmOperator

Date of creation

2013-03-22 14:17:00

Last modified on

2013-03-22 14:17:00

Owner

mhale (572)

Last modified by

mhale (572)

Numerical id

5

Author

mhale (572)

Entry type

Definition

Classification

msc 47A53

Generated on Fri Feb 9 21:45:42 2018 by

LaTeXML
Kato-Rellich theorem

Kato-Rellich theorem

Let

ℋ

be a

Hilbert space

,

A

:

D

⁢

(

A

)

⊂

ℋ

→

ℋ

a

self-adjoint operator

and

B

:

D

⁢

(

B

)

⊂

ℋ

→

ℋ

a symmetric operator with

D

⁢

(

A

)

⊂

D

⁢

(

B

)

.

We say that

B

is

A

-

bounded

if there are

positive

constants

α

,

β

such that

∥

B

⁢

x

∥

≤

α

⁢

∥

A

⁢

x

∥

+

β

⁢

∥

x

∥

for all

x

∈

D

⁢

(

A

)

, and we say that

α

is an

A

-bound

for

B

.

Theorem 1

.

(Kato-Rellich) If

B

is

A

-bounded

with

A

-bound smaller
than

1

, then

A

+

B

is

self-adjoint

on

D

⁢

(

A

)

, and essentially
self-adjoint on any core of

A

. Moreover, if

A

is bounded below, then
so is

A

+

B

.

Title

Kato-Rellich theorem

Canonical name

KatoRellichTheorem

Date of creation

2013-03-22 14:52:59

Last modified on

2013-03-22 14:52:59

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

7

Author

Koro (127)

Entry type

Theorem

Classification

msc 47A55

Synonym

Rellich-Kato theorem

Defines

A-bounded

Defines

A-bound

Generated on Fri Feb 9 21:45:44 2018 by

LaTeXML
proof of Taylor’s formula for matrix functions

proof of Taylor’s formula for matrix functions

Theorem

.

Let

p

be a

polynomial

and suppose

A

and

B

are squared matrices of the same size, then

p

⁢

(

A

+

B

)

=

∑

k

=

0

n

1

k

!

⁢

p

(

k

)

⁢

(

A

)

⁢

B

k

where

n

=

deg

⁡

(

p

)

.

Proof.

Since

p

is a polynomial, we can apply the

Taylor expansion

:

p

⁢

(

x

)

=

∑

k

=

0

n

1

k

!

⁢

p

(

k

)

⁢

(

x

0

)

⁢

(

x

-

x

0

)

k

where

n

=

deg

⁡

(

p

)

. Now let

x

=

𝐀

+

𝐁

and

x

0

=

𝐀

.

The Taylor expansion can be checked as follows: let

p

⁢

(

x

)

=

∑

k

=

0

n

a

k

⁢

x

k

for coefficients

a

k

(note that this coefficients can be taken from the space of square
matrices defined over a field). We define the formal derivative of
this polynomial as

p

(

1

)

⁢

(

x

)

=

d

⁢

p

d

⁢

x

=

∑

k

=

1

n

a

k

⁢

k

⁢

x

k

-

1

and we define

p

(

k

)

=

d

⁢

p

(

k

-

1

)

d

⁢

x

.

Then

p

(

k

)

⁢

(

x

)

=

∑

i

=

k

n

a

i

⁢

i

!

(

i

-

k

)

!

⁢

x

i

-

k

and we have

1

k

!

⁢

p

(

k

)

⁢

(

x

0

)

=

∑

i

=

k

n

a

i

⁢

i

!

(

i

-

k

)

!

⁢

k

!

⁢

(

x

0

)

i

-

k

.
Now consider

∑

k

=

0

n

1

k

!

⁢

p

(

k

)

⁢

(

x

0

)

⁢

(

x

-

x

0

)

k

=

∑

k

=

0

n

(

∑

i

=

k

n

a

i

⁢

i

!

(

i

-

k

)

!

⁢

k

!

⁢

(

x

0

)

i

-

k

⁢

(

x

-

x

0

)

k

)

=

∑

i

=

0

n

a

i

⁢

(

x

0

)

i

+

∑

i

=

1

n

a

i

⁢

i

⁢

(

x

0

)

i

-

1

⁢

(

x

-

x

0

)

+

…

+

∑

i

=

j

n

a

i

⁢

i

!

(

i

-

j

)

!

⁢

j

!

⁢

(

x

0

)

i

-

j

⁢

(

x

-

x

0

)

j

+

…

+

a

n

⁢

(

x

-

x

0

)

n

=

a

0

+

a

1

⁢

(

x

)

+

…

+

a

i

⁢

(

∑

j

=

0

i

i

!

(

i

-

j

)

!

⁢

j

!

⁢

(

x

0

)

i

-

j

⁢

(

x

-

x

0

)

j

)

+

…

+

a

n

⁢

(

∑

j

=

0

n

n

!

(

n

-

j

)

!

⁢

j

!

⁢

(

x

0

)

n

-

j

⁢

(

x

-

x

0

)

j

)

=

∑

k

=

0

n

a

k

⁢

x

i

=

p

⁢

(

x

)

since

∑

j

=

0

i

i

!

(

i

-

j

)

!

⁢

j

!

⁢

(

x

0

)

i

-

j

⁢

(

x

-

x

0

)

j

=

(

x

)

i

.
∎

Title

proof of Taylor’s formula for matrix functions

Canonical name

ProofOfTaylorsFormulaForMatrixFunctions

Date of creation

2013-03-22 17:57:04

Last modified on

2013-03-22 17:57:04

Owner

joen235 (18354)

Last modified by

joen235 (18354)

Numerical id

4

Author

joen235 (18354)

Entry type

Proof

Classification

msc 47A56

Generated on Fri Feb 9 21:45:49 2018 by

LaTeXML
Taylor’s formula for matrix functions

Taylor’s formula for matrix functions

Let

p

be a

polynomial

and suppose

𝐀

and

𝐁

commute, i.e.

𝐀𝐁

=

𝐁𝐀

, then

p

⁢

(

𝐀

+

𝐁

)

=

∑

k

=

0

n

1

k

!

⁢

p

(

k

)

⁢

(

𝐀

)

⁢

𝐁

k

.

where

n

=

deg

⁡

(

p

)

.

Title

Taylor’s formula for matrix functions

Canonical name

TaylorsFormulaForMatrixFunctions

Date of creation

2013-03-22 13:39:25

Last modified on

2013-03-22 13:39:25

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

7

Author

bwebste (988)

Entry type

Theorem

Classification

msc 47A56

Generated on Fri Feb 9 21:45:55 2018 by

LaTeXML
vector measure

vector measure

Let

S

be a set and

ℱ

a

field of sets

of

S

. Let

X

be a

topological vector space

.

A

vector measure

is a

function

μ

:

ℱ

⟶

X

that is , i.e. for any two

disjoint sets

A

1

,

A

2

in

ℱ

we have

μ

⁢

(

A

1

∪

A

2

)

=

μ

⁢

(

A

1

)

+

μ

⁢

(

A

2

)

A vector measure

μ

is said to be if for any

sequence

(

A

n

)

n

∈

ℕ

of disjoint sets in

ℱ

such that

⋃

n

=

1

∞

A

n

∈

ℱ

one has

μ

⁢

(

⋃

n

=

1

∞

A

n

)

=

∑

n

=

1

∞

μ

⁢

(

A

n

)

where the series

converges

in the topology of

X

.

In the particular case when

X

=

ℂ

, a countably additive vector measure is usually called a

complex measure

.

Thus, vector measures are to

measures

and

signed measures

but they take values on a vector space (with a particular topology).

0.0.1

Examples :

•

Let

(

X

,

𝔅

,

λ

)

be a

measure space

. Consider the

Banach space

L

p

⁢

(

X

,

𝔅

,

λ

)

(

http://planetmath.org/LpSpace

) with

1

≤

p

≤

∞

. Define the the function

μ

:

𝔅

⟶

L

p

⁢

(

X

,

𝔅

,

μ

)

by

μ

⁢

(

A

)

:=

χ

A

where

χ

A

denotes the

characteristic function

of the

measurable set

A

. It is easily seen that

μ

is a vector measure, which is

countably additive

if

1

≤

p

<

∞

(in case

p

=

∞

, countably additiveness fails).

•

spectral measures

are vector measures in the

σ

-algebra of

Borel sets

in

ℂ

whose values are projections on some

Hilbert space

. They are used in general formulations of the

spectral theorem

.

Title

vector measure

Canonical name

VectorMeasure

Date of creation

2013-03-22 17:29:23

Last modified on

2013-03-22 17:29:23

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

12

Author

asteroid (17536)

Entry type

Definition

Classification

msc 47A56

Classification

msc 46G12

Classification

msc 46G10

Classification

msc 28C20

Classification

msc 28B05

Defines

complex measure

Defines

countably additive vector measure

Generated on Fri Feb 9 21:45:57 2018 by

LaTeXML
Beltrami identity

Beltrami identity

Let

q

⁢

(

t

)

be a function

ℝ

→

ℝ

,

q

˙

=

d

d

⁢

t

⁢

q

, and

L

=

L

⁢

(

q

,

q

˙

,

t

)

. Begin with the time-relative

Euler-Lagrange condition

∂

∂

⁡

q

⁢

L

-

d

d

⁢

t

⁢

(

∂

∂

⁡

q

˙

⁢

L

)

=

0

.

(1)

If

∂

∂

⁡

t

⁢

L

=

0

, then the Euler-Lagrange condition reduces to

L

-

q

˙

⁢

∂

∂

⁡

q

˙

⁢

L

=

C

,

(2)

which is the

Beltrami identity

. In the

calculus of variations

, the ability to use the Beltrami identity can vastly simplify problems, and as it happens, many physical problems have

∂

∂

⁡

t

⁢

L

=

0

.

In space-relative terms, with

q

′

:=

d

d

⁢

x

⁢

q

, we have

∂

∂

⁡

q

⁢

L

-

d

d

⁢

x

⁢

∂

∂

⁡

q

′

⁢

L

=

0

.

(3)

If

∂

∂

⁡

x

⁢

L

=

0

, then the Euler-Lagrange condition reduces to

L

-

q

′

⁢

∂

∂

⁡

q

′

⁢

L

=

C

.

(4)

To derive the Beltrami identity, note that

d

d

⁢

t

⁢

(

q

˙

⁢

∂

∂

⁡

q

˙

⁢

L

)

=

q

¨

⁢

∂

∂

⁡

q

˙

⁢

L

+

q

˙

⁢

d

d

⁢

t

⁢

(

∂

∂

⁡

q

˙

⁢

L

)

(5)

Multiplying (1) by

q

˙

, we have

q

˙

⁢

∂

∂

⁡

q

⁢

L

-

q

˙

⁢

d

d

⁢

t

⁢

(

∂

∂

⁡

q

˙

⁢

L

)

=

0

.

(6)

Now, rearranging (5) and substituting in for the rightmost term of (6), we obtain

q

˙

⁢

∂

∂

⁡

q

⁢

L

+

q

¨

⁢

∂

∂

⁡

q

˙

⁢

L

-

d

d

⁢

t

⁢

(

q

˙

⁢

∂

∂

⁡

q

˙

⁢

L

)

=

0

.

(7)

Now consider the total derivative

d

d

⁢

t

⁢

L

⁢

(

q

,

q

˙

,

t

)

=

q

˙

⁢

∂

∂

⁡

q

⁢

L

+

q

¨

⁢

∂

∂

⁡

q

˙

⁢

L

+

∂

∂

⁡

t

⁢

L

.

(8)

If

∂

∂

⁡

t

⁢

L

=

0

, then we can substitute in the left-hand side of (8) for the leading portion of (7) to get

d

d

⁢

t

⁢

L

-

d

d

⁢

t

⁢

(

q

˙

⁢

∂

∂

⁡

q

˙

⁢

L

)

=

0

.

(9)

Integrating with respect to

t

, we arrive at

L

-

q

˙

⁢

∂

∂

⁡

q

˙

⁢

L

=

C

,

(10)

which is the Beltrami identity.

Title

Beltrami identity

Canonical name

BeltramiIdentity

Date of creation

2013-03-22 12:21:08

Last modified on

2013-03-22 12:21:08

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 47A60

Related topic

CalculusOfVariations

Related topic

EulerLagrangeDifferentialEquation

Generated on Fri Feb 9 21:45:59 2018 by

LaTeXML
C^*-algebra homomorphisms preserve continuous functional calculus

C

*

-algebra homomorphisms preserve continuous functional calculus

Let us setup some notation first: Let

𝒜

be a unital

C

*

-

algebra

(

http://planetmath.org/CAlgebra

) and

z

a

normal element

of

𝒜

. Then

•

σ

⁢

(

z

)

denotes the

spectrum

of

z

.

•

C

⁢

(

σ

⁢

(

z

)

)

denotes the

C

*

-algebra of

continuous functions

σ

⁢

(

z

)

⟶

ℂ

.

•

If

f

∈

C

⁢

(

σ

⁢

(

z

)

)

then

f

⁢

(

z

)

is the element of

𝒜

given by the

continuous functional calculus

.

Theorem

-

Let

𝒜

,

ℬ

be unital

C

*

-algebras (

http://planetmath.org/CAlgebra

) and

Φ

:

𝒜

⟶

ℬ

a *-

homomorphism

. Let

x

be a normal element in

𝒜

. If

f

∈

C

⁢

(

σ

⁢

(

x

)

)

then

Φ

⁢

(

f

⁢

(

x

)

)

=

f

⁢

(

Φ

⁢

(

x

)

)

Proof:

The

identity elements

of

𝒜

and

ℬ

will be both denoted by

e

and it will be clear from the

context

which one we are referring to.

First, we need to check that

f

⁢

(

Φ

⁢

(

x

)

)

is a

well-defined

element of

ℬ

, i.e. that

σ

⁢

(

Φ

⁢

(

x

)

)

⊆

σ

⁢

(

x

)

. This is clear since, if

x

-

λ

⁢

e

is

invertible

for some

λ

∈

ℂ

, then

Φ

⁢

(

x

)

-

λ

⁢

e

=

Φ

⁢

(

x

-

λ

⁢

e

)

is also invertible.

Let

{

p

n

}

be

sequence

of

polynomials

in

C

⁢

(

σ

⁢

(

x

)

)

converging uniformly to

f

. Then we have that

•

Φ

⁢

(

p

n

⁢

(

x

)

)

⟶

Φ

⁢

(

f

⁢

(

x

)

)

, by the

continuity

of

Φ

(see this entry (

http://planetmath.org/HomomorphismsOfCAlgebrasAreContinuous

)) and the continuity of the continuous functional calculus

mapping

.

•

p

n

⁢

(

Φ

⁢

(

x

)

)

⟶

f

⁢

(

Φ

⁢

(

x

)

)

, by the continuity of the continuous functional calculus mapping.

It is easily checked that

Φ

⁢

(

p

n

⁢

(

x

)

)

=

p

n

⁢

(

Φ

⁢

(

x

)

)

(since

Φ

is an homomorphism). Hence we conclude that

Φ

⁢

(

f

⁢

(

x

)

)

=

f

⁢

(

Φ

⁢

(

x

)

)

as intended.

□

Title

C

*

-algebra homomorphisms

preserve

continuous functional calculus

Canonical

name

CalgebraHomomorphismsPreserveContinuousFunctionalCalculus

Date of creation

2013-03-22 18:00:50

Last modified on

2013-03-22 18:00:50

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 47A60

Classification

msc 46L05

Generated on Fri Feb 9 21:46:13 2018 by

LaTeXML
classical isoperimetric problem

classical isoperimetric problem

The points

a

and

b

on the

x

-axis have to be by an arc with a given length (

http://planetmath.org/ArcLength

)

l

such that the area between the

x

-axis and the arc is as great as possible.

Denote the equation of the searched arc by

y

=

y

⁢

(

x

)

.  The task, which belongs to the

isoperimetric problems

(

http://planetmath.org/IsoperimetricProblem

), can be formulated as

to maximise

∫

a

b

y

⁢

𝑑

x

(1)

under the constraint condition

∫

a

b

1

+

y

′

⁣

2

⁢

𝑑

x

=

l

.

(2)

We have the integrands

f

⁢

(

x

,

y

,

y

′

)

≡

y

,

g

⁢

(

x

,

y

,

y

′

)

≡

1

+

y

′

⁣

2

.

The

variation

problem for the functional in (1) may be considered as a free variation problem (without conditions) for the functional

∫

a

b

(

f

-

λ

⁢

g

)

⁢

𝑑

x

where

λ

is a

Lagrange multiplier

.  For this end we need the Euler–Lagrange

differential equation

(

http://planetmath.org/EulerLagrangeDifferentialEquation

)

∂

∂

⁡

y

⁢

(

f

-

λ

⁢

g

)

-

d

d

⁢

x

⁢

∂

∂

⁡

y

′

⁢

(

f

-

λ

⁢

g

)

=

0

.

(3)

Since the expression

f

-

λ

⁢

g

does not depend explicitly on

x

, the differential equation (3) has, by the

Beltrami identity

, a first integral of the form

(

f

-

λ

⁢

g

)

-

y

′

⋅

(

f

y

′

′

-

λ

⁢

g

y

′

′

)

=

C

2

,

which reads simply

y

-

λ

1

+

y

′

⁣

2

=

C

2

.

This differential equation may be written

y

′

≡

d

⁢

y

d

⁢

x

=

λ

2

-

(

y

-

C

2

)

2

y

-

C

2

,

where one can separate the variables (

http://planetmath.org/SeparationOfVariables

) and integrate, obtaining the equation

(

x

-

C

1

)

2

+

(

y

-

C

2

)

2

=

λ

2

of a circle.  Here, the parametres

C

1

,

C

2

,

λ

may be determined from the conditions

y

⁢

(

a

)

=

y

⁢

(

b

)

=

0

,

arc length

=

l

.

Thus the

extremal

of this variational problem is a circular arc (

http://planetmath.org/CircularSegment

) connecting the given points.

Note that in every point of the arc, the angle of view of the line segment between the given points is constant.

Title

classical isoperimetric problem

Canonical name

ClassicalIsoperimetricProblem

Date of creation

2013-03-22 19:10:26

Last modified on

2013-03-22 19:10:26

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

22

Author

pahio (2872)

Entry type

Example

Classification

msc 47A60

Classification

msc 49K22

Classification

msc 49K05

Related topic

LagrangeMultiplierMethod

Related topic

CircularSegment

Related topic

AngleOfViewOfALineSegment

Generated on Fri Feb 9 21:46:23 2018 by

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derivation of Euler-Lagrange differential equation (advanced)

derivation of Euler-Lagrange differential equation (advanced)

Suppose that

x

0

∈

D

. Choose

r

such that the closed ball of radius

r

about

x

0

is contained in

D

. Let

q

be any function whose support lies in this closed ball.

By the definition of

F

,

∂

∂

⁡

λ

⁢

F

⁢

(

q

0

+

λ

⁢

q

)

=

∂

∂

⁡

λ

⁢

∫

D

L

⁢

(

x

,

q

0

+

λ

⁢

q

,

d

⁢

q

0

+

λ

⁢

d

⁢

q

)

⁢

d

m

⁢

x

=

∂

∂

⁡

λ

⁢

(

∫

|

x

-

x

0

|

≤

r

L

⁢

(

x

,

q

0

+

λ

⁢

q

,

d

⁢

q

0

+

λ

⁢

d

⁢

q

)

⁢

d

m

⁢

x

+

∫

x

∈

D

|

x

-

x

0

|

>

r

L

⁢

(

x

,

q

0

+

λ

⁢

q

,

d

⁢

q

0

+

λ

⁢

d

⁢

q

)

⁢

d

m

⁢

x

)

By the condition imposed on

q

, the

derivative

of the second

integral

is zero. Since the integrand of the first integral and its

first derivatives

are

continuous

and the closed ball is

compact

, the integrand and its first derivatives are

uniformly continuous

, so it is permissible to interchange

differentiation

and integration. Hence,

∂

∂

⁡

λ

⁢

F

⁢

(

q

0

+

λ

⁢

q

)

=

∫

|

x

-

x

0

|

≤

r

∂

⁡

L

⁢

(

x

,

q

0

+

λ

⁢

q

,

d

⁢

q

0

+

λ

⁢

d

⁢

q

)

∂

⁡

λ

⁢

d

m

⁢

x

Title

derivation

of

Euler-Lagrange differential equation

(advanced)

Canonical name

DerivationOfEulerLagrangeDifferentialEquationadvanced

Date of creation

2013-03-22 14:45:13

Last modified on

2013-03-22 14:45:13

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Derivation

Classification

msc 47A60

Generated on Fri Feb 9 21:46:29 2018 by

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extremal

extremal

An extremum point (i.e. a maximum point or a minimum point) gives a maximum value or a minimum value for a

function

(

http://planetmath.org/RealFunction

).

An

extremal curve

(i.e. an

extremal

, or the corresponding

extremal function

) gives a maximum value or a minimum value for a

functional

(

http://planetmath.org/Functional

).

For example, if the functional is the length of a curve connecting two points on a sphere, the extremals are great circle arcs between the points.

Title

extremal

Canonical name

Extremal

Date of creation

2013-03-22 19:10:29

Last modified on

2013-03-22 19:10:29

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

4

Author

pahio (2872)

Entry type

Definition

Classification

msc 47A60

Synonym

extremal curve

Related topic

CalculusOfVariations

Defines

extremal function

Generated on Fri Feb 9 21:46:38 2018 by

LaTeXML
isoperimetric problem

isoperimetric problem

The simplest of the

isoperimetric problems

is the following:

One must set an arc with a given length

l

from a given point

P

of the plane to another given point

Q

such that the arc together with the line segment

P

⁢

Q

encloses the greatest area possible.

This task is solved in the entry example of

calculus of variations

.

More generally,

isoperimetric problem

may determining such an arc

c

between the given points

P

and

Q

that it gives for the integral

∫

P

Q

f

⁢

(

x

,

y

,

y

′

)

⁢

𝑑

s

(1)

an extremum and that gives for another integral

∫

P

Q

g

⁢

(

x

,

y

,

y

′

)

⁢

𝑑

s

(2)

a given value

l

, as both integrals are taken along

c

.  Here,

f

and

g

are given functions.

The constraint (2) can be omitted by using the function

f

-

λ

⁢

g

instead of

f

in (1) similarly as in the mentionned example.

Title

isoperimetric problem

Canonical name

IsoperimetricProblem

Date of creation

2013-03-22 19:12:01

Last modified on

2013-03-22 19:12:01

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Definition

Classification

msc 47A60

Classification

msc 49K22

Related topic

IsoperimetricInequality

Related topic

LagrangeMultiplier

Generated on Fri Feb 9 21:46:42 2018 by

LaTeXML
Russo-Dye theorem

Russo-Dye theorem

Let

A

be a unital

C

*

-algebra. Then the unit ball of

A

is the closed

convex hull

of the

unitaries

in

A

. Moreover, these

unitary elements

are

extreme points

of the unit ball.

Title

Russo-Dye theorem

Canonical name

RussoDyeTheorem

Date of creation

2013-03-22 16:01:52

Last modified on

2013-03-22 16:01:52

Owner

aube (13953)

Last modified by

aube (13953)

Numerical id

4

Author

aube (13953)

Entry type

Theorem

Classification

msc 47A60

Generated on Fri Feb 9 21:46:52 2018 by

LaTeXML
stationary point

stationary point

Suppose

V

is a

vector space

and

L

:

V

→

ℝ

is a map.
Then

v

∈

V

is a

stationary point

of

L

provided that

d

d

⁢

t

⁢

L

⁢

(

v

+

t

⁢

u

)

|

t

=

0

=

0

for all

u

∈

V

. In this case

u

is called a variation of

v

.

Title

stationary point

Canonical name

StationaryPoint

Date of creation

2013-03-22 15:33:21

Last modified on

2013-03-22 15:33:21

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Definition

Classification

msc 47A60

Generated on Fri Feb 9 21:46:54 2018 by

LaTeXML
eigenvalues of normal operators

eigenvalues of normal operators

Let

H

be a

Hilbert space

and

B

⁢

(

H

)

the algebra of

bounded operators

in

H

. Suppose

T

∈

B

⁢

(

H

)

is a

normal operator

. Then

1.

- If

λ

∈

ℂ

is an

eigenvalue

of

T

, then

λ

¯

is an eigenvalue of

T

*

(the

adjoint operator

of

T

) for the same

eigenvector

.

2.

- Eigenvectors of

T

associated with distinct eigenvalues are

orthogonal

.

Remark -

It is known that for any

linear operator

eigenvectors associated with distinct eigenvalues are

linearly independent

. 2 strengthens this result for normal operators.

Title

eigenvalues of normal operators

Canonical name

EigenvaluesOfNormalOperators

Date of creation

2013-03-22 17:33:32

Last modified on

2013-03-22 17:33:32

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

10

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 47B15

Classification

msc 47A75

Classification

msc 47A15

Classification

msc 47A10

Classification

msc 15A18

Generated on Fri Feb 9 21:46:58 2018 by

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self-adjoint operator

self-adjoint operator

A

densely defined

linear operator

A

:

𝒟

⁢

(

A

)

⊂

ℋ

→

ℋ

on a Hilbert space

ℋ

is a

Hermitian

or

symmetric

operator

if

(

A

⁢

x

,

y

)

=

(

x

,

A

⁢

y

)

for all

x

,

y

∈

𝒟

⁢

(

A

)

. This means that the

adjoint

A

*

of

A

is defined at least on

𝒟

⁢

(

A

)

and that its

restriction

to that set coincides with

A

. This fact is often denoted by

A

⊂

A

*

.

The operator

A

is

self-adjoint

if it coincides with its adjoint, i.e. if

A

=

A

*

.
If

A

is

closable

and its

closure

coincides with its adjoint (i.e.

A

¯

=

A

*

), then

A

is said to be

essentially self-adjoint

.

Title

self-adjoint operator

Canonical name

SelfadjointOperator

Date of creation

2013-03-22 13:48:23

Last modified on

2013-03-22 13:48:23

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Definition

Classification

msc 47B15

Classification

msc 47B25

Related topic

HermitianMatrix

Defines

Hermitian operator

Defines

symmetric operator

Defines

essentially self-adjoint

Defines

self-adjoint

Generated on Fri Feb 9 21:47:00 2018 by

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