spectrum is a non-empty compact set

spectrum is a non-empty compact set

Theorem

-

Let

𝒜

be a complex

Banach algebra

with identity element. The spectrum of each

a

∈

𝒜

is a non-empty

compact set

in

ℂ

.

Remark

: For Banach algebras over

ℝ

the spectrum of an

element

is also a compact set, although it can be empty. To assure that it is not the

empty set

, proofs usually involve Liouville’s theorem (

http://planetmath.org/LiouvillesTheorem2

) for of a complex with values in a Banach algebra.

Proof :

Let

e

be the identity element of

𝒜

. Let

σ

⁢

(

a

)

denote the spectrum of the element

a

∈

𝒜

.

•

-

For each

λ

∈

ℂ

such that

|

λ

|

>

∥

a

∥

one has

∥

λ

-

1

⁢

a

∥

<

1

, and so, by the Neumann series (

http://planetmath.org/NeumannSeriesInBanachAlgebras

),

e

-

λ

-

1

⁢

a

is

invertible

. Since

a

-

λ

⁢

e

=

-

λ

⁢

(

e

-

λ

-

1

⁢

a

)

we see that

a

-

λ

⁢

e

is also invertible.

We conclude that

σ

⁢

(

a

)

is

contained

in a disk of radius

∥

a

∥

, and therefore it is

bounded

.

Let

ϕ

:

ℂ

⟶

𝒜

be the

function

defined by

ϕ

⁢

(

λ

)

=

a

-

λ

⁢

e

It is known that the set

𝒢

of the invertible elements of

𝒜

is open (see this entry (

http://planetmath.org/InvertibleElementsInABanachAlgebraFormAnOpenSet

)).

Since

ϕ

-

1

⁢

(

𝒢

)

=

ℂ

-

σ

⁢

(

a

)

and

ϕ

is a

continuous function

we see that that

σ

⁢

(

a

)

is a

closed set

in

ℂ

.

As

σ

⁢

(

a

)

is a bounded closed subset of

ℂ

, it is

compact

.

•

Non-emptiness -

Suppose that

σ

⁢

(

a

)

was empty. Then the resolvent

R

a

is defined in

ℂ

.

We can see that

R

a

is bounded since it is

continuous

in the closed disk

|

λ

|

<

∥

a

∥

and, for

λ

>

∥

a

∥

, we have (again, by the Neumann series (

http://planetmath.org/NeumannSeriesInBanachAlgebras

))

∥

R

a

⁢

(

λ

)

∥

=

∥

(

a

-

λ

⁢

e

)

-

1

∥

=

∥

λ

-

1

⁢

(

e

-

λ

-

1

⁢

a

)

-

1

∥

≤

|

λ

|

-

1

1

-

|

λ

|

-

1

⁢

∥

a

∥

=

1

|

λ

|

-

∥

a

∥

and therefore

lim

|

λ

|

→

∞

⁡

R

a

⁢

(

λ

)

=

0

, which shows that

R

a

is bounded.

The resolvent function,

R

a

, is

analytic

(

http://planetmath.org/BanachSpaceValuedAnalyticFunctions

) (see this entry (

http://planetmath.org/ResolventFunctionIsAnalytic

)). As it is defined in

ℂ

, it is a bounded entire function. Applying Liouville’s theorem (

http://planetmath.org/LiouvillesTheorem2

) we conclude that it must be

constant

(see this this entry (

http://planetmath.org/BanachSpaceValuedAnalyticFunctions

) for an idea of how
holds for

Banach space

valued functions).

Since

R

a

⁢

(

λ

)

converges

to

0

as

|

λ

|

→

∞

we see that

R

a

must be identically zero.

Thus, we have arrived to a

contradiction

since

0

is not invertible.

Therefore

σ

⁢

(

a

)

is non-empty.

□

Title

spectrum is a non-empty compact set

Canonical name

SpectrumIsANonemptyCompactSet

Date of creation

2013-03-22 17:25:05

Last modified on

2013-03-22 17:25:05

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

10

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 46H05

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