functional calculus

functional calculus

1

Basic Idea

Let

X

be a

normed vector space

over a field

𝕂

. Let

T

be a

linear operator

in

X

and

I

the

identity operator

in

X

.

The

functional calculus

refers to a specific process which enables the expression

f

⁢

(

T

)

to make sense as a linear operator in

X

, for certain

scalar functions

f

:

𝕂

⟶

𝕂

.

At first sight, and for most

functions

f

, there is no reason why the above expression should be associated with a particular linear operator.

But, for example, when

f

is a

polynomial

f

⁢

(

x

)

=

a

k

⁢

x

k

+

…

+

a

2

⁢

x

2

+

a

1

⁢

x

+

a

0

, the expression

f

⁢

(

T

)

:=

a

k

⁢

T

k

+

…

+

a

2

⁢

T

2

+

a

1

⁢

T

+

a

0

⁢

I

does indeed refer to a linear operator in

X

.

As another example, when

T

is a matrix in

ℝ

n

or

ℂ

n

one is sometimes led to the

exponential

(

http://planetmath.org/MatrixExponential

) of

T

e

T

=

∑

k

=

0

∞

T

k

k

!

Thus, we are applying the scalar

exponential function

to a matrix.

Note in this last example that

e

T

is approximated by polynomials (the partial sums of the series). This provides an idea of how to make sense of

f

⁢

(

T

)

if

f

can be approximated by polynomials:

If

f

can be approximated by polynomials

p

n

then one could try to define

f

⁢

(

T

)

:=

lim

n

→

∞

⁡

p

n

⁢

(

T

)

But for that one needs to define what “approximated” means and to assure the above limit exists.

2

More abstractly

There is no reason why one should restrict to linear operators in a normed vector space. In this , we can consider instead a unital

topological algebra

𝒜

over a field

𝕂

.

There is no definition in mathematics of

functional calculus

, but the ideas above show that a

functional calculus for an

element

T

∈

𝒜

should be something like an

homomorphism

(

⋅

)

⁢

(

T

)

:

ℱ

⟶

𝒜

from some topological algebra of scalar functions

ℱ

to

𝒜

, that satisfied the following :

•

ℱ

must contain the

polynomial functions

.

•

(

⋅

)

⁢

(

T

)

is

continuous

.

•

p

⁢

(

T

)

=

a

k

⁢

T

k

+

…

+

a

2

⁢

T

2

+

a

1

⁢

T

+

a

0

⁢

I

for each polynomial

p

⁢

(

x

)

=

a

k

⁢

x

k

+

…

+

a

2

⁢

x

2

+

a

1

⁢

x

+

a

0

, where

I

denotes the

identity element

of

𝒜

.

3

Functional Calculi

There are some

functional

calculi of . We give a very brief descprition of each one of them (please follows the links for entries with more detailed explanation).

•

polynomial functional calculus

-

This is valid for any element

T

in any

algebra

𝒜

. It

associates

polynomials to elements in the algebra generated by

T

, as discussed above.

•

functional calculus -

This is valid for any element

T

in a complex

Banach algebra

𝒜

. It associates

complex analytic functions

defined on the

spectrum

(

http://planetmath.org/Spectrum

) of

T

to elements in the algebra generated by

T

.

•

continuous functional calculus

-

This is valid for

normal elements

in

C

*

-algebras (

http://planetmath.org/CAlgebra

). It associates

continuous functions

on the spectrum of

T

to elements in the

C

*

-algebra generated by

T

.

•

Borel functional calculus

-

This is valid for

normal operators

T

in a

von Neumann algebra

𝒜

. It associates

bounded

Borel measurable functions on the spectrum of

T

to elements in the von Neumann algebra generated by

T

.

•

functional calculus for Hermitian matrices

-

A case. It is valid for

Hermitian matrices

T

. It associates real valued functions on the spectrum of

T

to elements in the algebra generated by

T

.

4

Applications

•

Functional calculi provide an of constructing new linear operators having specified out of given ones.

•

There are strong with

spectral theory

since one usually has

f

⁢

(

σ

⁢

(

T

)

)

=

σ

⁢

(

f

⁢

(

T

)

)

, where

σ

⁢

(

⋅

)

denotes the spectrum of its . This is called the

spectral mapping theorem

.

•

As the with spectral theory can possibly show, functional calculi are an tool for studying

equations

. For example, they can give

sufficient

conditions for the

existence

of a

square root

T

of an

T

.

Title

functional calculus

Canonical name

FunctionalCalculus

Date of creation

2013-03-22 17:29:40

Last modified on

2013-03-22 17:29:40

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

16

Author

asteroid (17536)

Entry type

Feature

Classification

msc 47A60

Classification

msc 46H30

Related topic

FunctionalCalculusForHermitianMatrices

Related topic

ContinuousFunctionalCalculus2

Related topic

PolynomialFunctionalCalculus

Related topic

BorelFunctionalCalculus

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