diagonalizable operator

diagonalizable operator

The expression ”

diagonalizable operator

” has several meanings in

operator theory

. The purpose of this entry is to present some commonly used

concepts

where this terminology appears.

0.1

Definition 1

Let

H

be a finite dimensional

Hilbert space

. A

linear operator

T

:

H

⟶

H

is said to be

diagonalizable

if the corresponding matrix (in a given basis) is a

diagonalizable matrix

(

http://planetmath.org/Diagonalizable2

).

The above definition is

equivalent

to: There exists a basis of

H

consisting of

eigenvectors

of

T

.

Remark -

This is a common definition in

linear algebra

.

0.2

Definition 2

Let

H

be a finite dimensional Hilbert space. A linear operator

T

:

H

⟶

H

is said to be

diagonalizable

if there is an

orthonormal basis

of

H

in which

T

is represented by a

diagonal matrix

.

The above definition is equivalent to: There exists an orthonormal basis of

H

consisting of eigenvectors of

T

.

Another equivalent definition is: There exists an orthonormal basis

{

e

1

,

…

,

e

n

}

of

H

and values

λ

1

,

…

,

λ

2

∈

ℂ

such that

T

⁢

(

x

)

=

∑

i

=

1

n

λ

i

⁢

⟨

x

,

e

i

⟩

⁢

e

i

Remarks -

•

In linear algebra (

http://planetmath.org/LinearAlgebra

) such

operators

are also called

unitarily diagonalizable

.

•

Diagonalizable operators (in this sense) are always

normal operators

. The

Spectral theorem

for normal operators (

http://planetmath.org/SpectralTheoremForHermitianMatrices

) assures that the

converse

is also true.

0.3

Definition 3

Let

H

be a Hilbert space. A

bounded linear operator

T

:

H

⟶

H

is said to be

diagonalizable

if there exists an orthonormal basis consisting of eigenvectors of

T

.

An equivalent definition is: There exists an orthonormal basis

{

e

i

}

i

∈

J

of

H

and values

{

λ

i

}

i

∈

J

such that

T

⁢

(

x

)

=

∑

i

∈

J

λ

i

⁢

⟨

x

,

e

i

⟩

⁢

e

i

Remarks -

•

If

H

is finite dimensional this is the same as definition 2.

•

Diagonalizable operators (in this sense) are always normal operators. For

compact operators

the converse is assured by an appropriate version of the spectral theorem for

compact

normal operators.

0.4

Definition 4

Let

H

be a Hilbert space. A linear operator

T

:

H

⟶

H

is said to be

diagonalizable

if it is to a

multiplication operator

(

http://planetmath.org/MultiplicationOperatorOnMathbbL22

) in some

L

2

-space (

http://planetmath.org/L2SpacesAreHilbertSpaces

), i.e. if there exists

•

a

measure space

(

X

,

ℬ

,

μ

)

,

•

a

unitary operator

U

:

L

2

⁢

(

X

)

⟶

H

and

•

a

function

f

∈

L

∞

⁢

(

X

)

(

http://planetmath.org/LpSpace

) such that

T

=

U

⁢

M

f

⁢

U

*

where

M

f

:

L

2

⁢

(

X

)

⟶

L

2

⁢

(

X

)

is the operator of

multiplication

(

http://planetmath.org/MultiplicationOperator

) by

f

M

f

⁢

(

ψ

)

=

f

.

ψ

.

Remarks -

•

If

H

=

ℂ

n

the above definition is equivalent to say that

T

is unitarily diagonalizable (Definition 2). Indeed, we can think of

ℂ

n

as

L

2

⁢

(

{

1

,

…

,

n

}

)

with the

counting measure

. In this case, multiplication operators correspond to diagonal matrices.

•

Diagonalizable operators (in this sense) are necessarily normal operators (since multiplication operators are so). The discussion about the converse result is the content of general versions of the spectral theorem.

Title

diagonalizable operator

Canonical name

DiagonalizableOperator

Date of creation

2013-03-22 17:33:47

Last modified on

2013-03-22 17:33:47

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Definition

Classification

msc 46C05

Classification

msc 47A05

Related topic

SpectralTheoremForHermitianMatrices

Defines

unitarily diagonalizable

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