polynomial functional calculus

polynomial functional calculus

Let

𝒜

be an unital

associative algebra

over

ℂ

with

identity element

e

and let

a

∈

𝒜

.

The

polynomial functional calculus

is the most basic form of a

functional calculus

. It allows the

expression

p

⁢

(

a

)

to make sense as an element of

𝒜

, for any

polynomial

p

:

ℂ

⟶

ℂ

.

This is achieved in the following natural way: for any polynomial

p

⁢

(

λ

)

:=

∑

c

n

⁢

λ

n

we the

element

p

⁢

(

a

)

:=

∑

c

n

⁢

a

n

∈

𝒜

.

1

Definition

Recall that the set of

polynomial functions

in

ℂ

, denoted by

ℂ

⁢

[

λ

]

, is an associative algebra over

ℂ

under

pointwise operations

and is generated by the

constant polynomial

1

and the

variable

λ

(corresponding to the

identity function

in

ℂ

).

Moreover, any

homomorphism

from the

algebra

ℂ

⁢

[

λ

]

is perfectly determined by the values of

1

and

λ

.

Definition

-

Consider the algebra homomorphism

π

:

C

⁢

[

λ

]

⟶

A

such that

π

⁢

(

1

)

=

e

and

π

⁢

(

λ

)

=

a

. This homomorphism is denoted by

p

⟼

p

⁢

(

a

)

and it is called the

polynomial functional calculus

for

a

.

It is clear that for any polynomial

p

⁢

(

λ

)

:=

∑

c

n

⁢

λ

n

we have

p

⁢

(

a

)

=

∑

c

n

⁢

a

n

.

2

Spectral Properties

We will denote by

σ

⁢

(

x

)

the spectrum (

http://planetmath.org/Spectrum

) of an element

x

∈

𝒜

.

Theorem

- (polynomial spectral mapping theorem) -

Let

A

be an unital associative algebra over

C

and

a

an element in

A

. For any polynomial

p

we have that

σ

⁢

(

p

⁢

(

a

)

)

=

p

⁢

(

σ

⁢

(

a

)

)

:

Let us first prove that

σ

⁢

(

p

⁢

(

a

)

)

⊆

p

⁢

(

σ

⁢

(

a

)

)

. Suppose

λ

~

∈

σ

⁢

(

p

⁢

(

a

)

)

, which means that

p

⁢

(

a

)

-

λ

~

⁢

e

is not

invertible

. Now consider the polynomial in

ℂ

given by

q

:=

p

-

λ

~

. It is clear that

q

⁢

(

a

)

=

p

⁢

(

a

)

-

λ

~

⁢

e

, and therefore

q

⁢

(

a

)

is not invertible. Since

ℂ

is

algebraically closed

(

http://planetmath.org/FundamentalTheoremOfAlgebra

), we have that

q

⁢

(

λ

)

=

(

λ

-

λ

1

)

n

1

⁢

⋯

⁢

(

λ

-

λ

k

)

n

k

for some

λ

1

,

…

,

λ

k

∈

ℂ

and

n

1

,

…

,

n

k

∈

ℕ

. Thus, we can also write a

similar

product

for

q

⁢

(

a

)

as

q

⁢

(

a

)

=

(

a

-

λ

1

⁢

e

)

n

1

⁢

⋯

⁢

(

a

-

λ

k

⁢

e

)

n

k

Now, since

q

⁢

(

a

)

is not invertible we must have that at least one of the factors

(

a

-

λ

i

⁢

e

)

is not invertible, which means that for that particular

λ

i

we have

λ

i

∈

σ

⁢

(

a

)

. But we also have that

q

⁢

(

λ

i

)

=

0

, i.e.

p

⁢

(

λ

i

)

=

λ

~

, and hence

λ

~

∈

p

⁢

(

σ

⁢

(

a

)

)

.

We now prove the

inclusion

σ

⁢

(

p

⁢

(

a

)

)

⊇

p

⁢

(

σ

⁢

(

a

)

)

. Suppose

λ

~

∈

p

⁢

(

σ

⁢

(

a

)

)

, which means that

λ

~

=

p

⁢

(

λ

0

)

for some

λ

0

∈

σ

⁢

(

a

)

. The polynomial

p

-

λ

~

has a zero at

λ

0

, hence there is a polynomial

d

such that

p

⁢

(

λ

)

-

λ

~

=

d

⁢

(

λ

)

⁢

(

λ

-

λ

0

)

,

λ

∈

ℂ

Thus, we can also write a similar product for

q

⁢

(

a

)

as

p

⁢

(

a

)

-

λ

~

⁢

e

=

d

⁢

(

a

)

⁢

(

a

-

λ

0

⁢

e

)

If

p

⁢

(

a

)

-

λ

~

⁢

e

was invertible, then we would see that

a

-

λ

0

⁢

e

had a left (

http://planetmath.org/InversesInRings

) and a

right inverse

(

http://planetmath.org/InversesInRings

), thus being invertible. But we know that

λ

0

∈

σ

⁢

(

a

)

, hence we conclude that

p

⁢

(

a

)

-

λ

~

⁢

e

cannot be invertible, i.e.

λ

~

∈

σ

⁢

(

p

⁢

(

a

)

)

.

□

Title

polynomial functional calculus

Canonical

name

PolynomialFunctionalCalculus

Date of creation

2013-03-22 18:48:23

Last modified on

2013-03-22 18:48:23

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

8

Author

asteroid (17536)

Entry type

Feature

Classification

msc 46H30

Classification

msc 47A60

Related topic

FunctionalCalculus

Related topic

ContinuousFunctionalCalculus2

Related topic

BorelFunctionalCalculus

Defines

polynomial spectral mapping theorem

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