spectral theorem

spectral theorem

\PMlinkescapephrase

theory

\PMlinkescapephrase

state

\PMlinkescapephrase

theory

\PMlinkescapephrase

series

\PMlinkescapephrase

bounded

\PMlinkescapephrase

unbounded

1

Introduction

The

spectral theorem

is series of results in

functional analysis

that explore conditions for

operators

in

Hilbert spaces

to be

diagonalizable

(in some appropriate sense). These results can also describe how the

diagonalization

takes , mainly by analyzing how the operator acts in the underlying Hilbert space.

Roughly speaking, the spectral theorems state that normal operators (or

self-adjoint operators

) are diagonalizable and can be expressed as a sum or, more generally, as an

integral

of

projections

. More specifically, a normal (or

self-adjoint

) operator

T

is

unitarily equivalent

to a

multiplication operator

in some

L

2

-space (

http://planetmath.org/L2SpacesAreHilbertSpaces

) and we can to it a

spectral measure

(or a resolution of identity) whose integration gives is

T

.

There is a wide of spectral theorems, each one with its own , applicable to many classes of normal and self-adjoint operators.

2

Motivation

We explore here two to motivate the spectral theorem. The first is by recalling the

finite-dimensional

case, corresponding to the well known result in

linear algebra

, the spectral theorem for

Hermitian matrices

(

http://planetmath.org/SpectralTheoremForHermitianMatrices

) (or the self-adjoint analog). The second motivation comes from the

C

*

-

algebra

(

http://planetmath.org/CAlgebra

) theory, by regarding a normal (or self-adjoint) operator as a

continuous function

.

2.0.1

Finite-dimensional case

Suppose

T

is a self-adjoint operator in a finite-dimensional Hilbert space

H

. An important fact about self-adjoint operators (not just in finite-dimensional spaces) is the following:

Fact 1 -

If

V

⊆

H

is an

invariant subspace

by

T

, then so it is

V

⟂

, the

orthogonal complement

of

V

.

Proof:

Let

x

∈

V

and

y

∈

V

⟂

. Then

⟨

x

,

T

⁢

y

⟩

=

⟨

T

⁢

x

,

y

⟩

=

0

, where the last term is zero because

V

is

invariant

by

T

, i.e.

T

⁢

x

∈

V

. But this proves that

T

⁢

y

∈

V

⟂

.

□

In finite

dimensions

it is known that every

linear transformation

has, at least, one

eigenvector

. Of course, the

subspace

generated by

an eigenvector is always invariant.

Let

v

1

be an eigenvector of

T

and

V

1

the subspace generated by it. For self-adjoint

transformations

, Fact 1 above says that

V

1

⟂

is also invariant by

T

. Thus, by

restriction

(

http://planetmath.org/RestrictionOfAFunction

), we have a self-adjoint operator

T

:

V

1

⟂

⟶

V

1

⟂

and we could again find an eigenvector and repeat the same . Thus, we are decomposing

H

as a

direct sum

of

orthogonal

one-dimensional subspaces

H

=

V

1

⊕

…

⊕

V

n

, and the operator

T

can be expressed as a sum

T

=

∑

i

=

1

n

λ

i

⁢

P

i

where each

λ

i

is the

eigenvalue

associated with the eigenvector

v

i

and each

P

i

is the orthogonal projection onto the subspace

V

i

.

This is exactly the process of diagonalization of a

self-adjoint matrix

.

For normal operators it is more subtle as Fact 1 is no longer true. The idea to overpass this is that eigenvectors of normal operators are always orthogonal to each other (see this entry (

http://planetmath.org/EigenvaluesOfNormalOperators

)).

2.0.2

C

*

-algebras

Suppose

T

is a self-adjoint operator in some Hilbert space

H

. The closed *-algebra generated by

T

and the

identity operator

is a

commutative

C

*

-algebra (

http://planetmath.org/CAlgebra

), which we denote by

C

*

⁢

(

T

)

. Hence, the

Gelfand-Naimark theorem

(

http://planetmath.org/GelfandTransform

) and the

continuous functional calculus

provide an

isomorphism

C

*

⁢

(

T

)

≅

C

⁢

(

σ

⁢

(

T

)

)

where

σ

⁢

(

T

)

stands for the

spectrum

of

T

and

C

⁢

(

σ

⁢

(

T

)

)

is the

C

*

-algebra of continuous functions

σ

⁢

(

T

)

→

ℂ

.

Recall that the spectrum of a self-adjoint operator is a always a

compact subset

of

ℝ

. Thus, we can think of

T

as a continuous function defined in a subset of

ℝ

.

It is a well known fact from

measure theory

that every continuous function

f

:

X

⟶

ℂ

can be approximated by

linear combinations

of

characteristic functions

. With some additional effort it can be seen that each continuous function

f

is in fact a (vector valued) integral of characteristic functions

f

=

∫

X

f

⁢

𝑑

χ

where

χ

is the

vector measure

of characteristic functions

χ

⁢

(

A

)

:=

χ

A

.

We now notice that characteristic functions in

σ

⁢

(

T

)

are not

continuous

in general. Hence, they may not have a correspondent in the

C

*

⁢

(

T

)

, the

C

*

-algebra generated by

T

and the

identity

. The fact is that they do have a correspondent in the

von Neumann algebra

generated by

T

. Informally, this is the same as saying that characteristic functions belong to

L

∞

⁢

(

σ

⁢

(

T

)

)

rather then

C

⁢

(

σ

⁢

(

T

)

)

.

The correspondent operators in the von Neumann algebra generated by

T

must be projections (since characteristic functions are projections in

L

∞

), and similarly,

T

can be approximated by linear combinations of projections and can, in fact, be expressed as an integral of projections:

T

=

∫

σ

⁢

(

T

)

λ

⁢

𝑑

P

⁢

(

λ

)

where

P

⁢

(

λ

)

is a resolution of identity for

T

(or a projection valued measure, when

T

is a normal operator).

3

Spectral Theorems

Here we list a series of spectral theorems, applicable to different classes of normal or self-adjoint operators.

3.0.1

Normal Operators

•

spectral theorem for

Hermitean

matrices (

http://planetmath.org/SpectralTheoremForHermitianMatrices

)

This is the finite-dimensional case. It is often referred to as ”the spectral theorem”, especially in linear algebra.

•

spectral theorem for bounded (

http://planetmath.org/OperatorNorm

) normal operators in separable Hilbert spaces

•

spectral theorem for bounded normal operators in inseparable Hilbert spaces

•

spectral theorem for

compact

(

http://planetmath.org/CompactOperator

) normal operators

•

spectral theorem for

unitary operators

•

spectral theorem for unbounded (

http://planetmath.org/OperatorNorm

) normal operators

3.0.2

Self-adjoint Operators

•

spectral theorem for self-adjoint matrices

•

spectral theorem for bounded self-adjoint operators in separable Hilbert spaces

•

spectral theorem for bounded self-adjoint operators in inseparable Hilbert spaces

•

spectral theorem for self-adjoint operators

•

spectral theorem for unbounded self-adjoint operators

Title

spectral theorem

Canonical name

SpectralTheorem1

Date of creation

2013-03-22 18:04:30

Last modified on

2013-03-22 18:04:30

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

12

Author

asteroid (17536)

Entry type

Feature

Classification

msc 47B15

Classification

msc 47A15

Classification

msc 47A10

Classification

msc 46C99

Classification

msc 15A18

Generated on Fri Feb 9 21:47:03 2018 by

LaTeXML