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functional calculus

(Feature)

Basic Idea

Let

be a normed vector space over a field $\mathbb{K}$ . Let

be a linear operator in

and

the identity operator in

The term functional calculus refers to a specific process which enables the expression

$\displaystyle f(T)$

to make sense as a linear operator in

, for certain scalar functions $f: \mathbb{K} \longrightarrow \mathbb{K}$ .

At first sight, and for most functions , there is no reason why the above expression should be associated with a particular linear operator.

But, for example, when is a polynomial $f(x)=a_kx^n + \dots + a_2x^2 + a_1x + a_0$ , the expression

$\displaystyle f(T):=a_kT^n + \dots + a_2T^2 + a_1T +a_0I$

does indeed refer to a linear operator in

As another example, when is a matrix in $\mathbb{R}^n$ or $\mathbb{C}^n$ one is sometimes led to the exponential of

$\displaystyle e^T=\sum_{k=0}^{\infty} \frac{T^k}{k!}$

Thus, we are applying the scalar exponential function to a matrix.

Note in this last example that is approximated by polynomials (the partial sums of the series). This provides an idea of how to make sense of if can be approximated by polynomials:

If can be approximated by polinomials then one could try to define

$\displaystyle f(T):=\lim_{n \rightarrow \infty} p_n(T)$

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

More abstractly

There is no reason why one should restrict to linear operators in a normed vector space. In this way, we can consider instead a unital topological algebra $\mathcal{A}$ over a field $\mathbb{K}$ .

There is no definition in mathematics of functional calculus, but the ideas above show that a functional calculus for an element $T \in \mathcal{A}$ should be something like an homomorphism $(\cdot)(T) : \mathcal{F} \longrightarrow \mathcal{A}$ from some topological algebra of scalar functions $\mathcal{F}$ to $\mathcal{A}$ , that satisfied the following properties:

$\mathcal{F}$ must contain the polynomial functions.
$(\cdot)(T)$ is continuous.
$p(T)= a_kT^n + \dots + a_2T^2 + a_1T +a_0I$ for each polynomial $p(x)=a_kx^n + \dots + a_2x^2 + a_1x + a_0$ , where denotes the identity element of $\mathcal{A}$ .

Functional Calculi

There are some functional calculi of interest. We give a very brief descprition of each one of them (the corresponding entries with more detailed explanation are still in preparation).

polynomial functional calculus -
This is valid for any element in any algebra $\mathcal{A}$ . It associates polynomials to elements in the algebra generated by , as discussed above.
analytic functional calculus -
This is valid for any element in a complex Banach algebra $\mathcal{A}$ . It associates complex analytic functions defined on the spectrum of to elements in the algebra generated by .
continuous functional calculus -
This is valid for normal elements in -algebras. It associates continuous functions on the spectrum of to elements in the algebra generated by .
borel functional calculus -
This is valid for normal operators in a von Neumann algebra $\mathcal{A}$ . It associates borel measurable functions on the spectrum of to elements in the algebra generated by .
functional calculus for Hermitian matrices -
A finite dimensional case. It is valid for Hermitian matrices . It associates real valued functions on the spectrum of to elements in the algebra generated by .

Applications

Functional calculi provide an effective way of constructing new linear operators having specified properties out of given ones.
There are strong connections with spectral theory since one usually has $f(\sigma(T)) = \sigma(f(T))$ , where $\sigma(\cdot)$ denotes the spectrum of its argument. This is called the spectral mapping theorem.
As the connections with spectral theory can possibly show, functional calculi are an effective tool for studying operator equations. For example, they can give sufficient conditions for the existence of a square root $\sqrt{T}$ of an operator .

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Cross-references: square root, sufficient, equations, spectral mapping theorem, real, Hermitian matrices, functional calculus for Hermitian matrices, Borel measurable functions, von Neumann algebra, normal operators, normal elements, continuous functional calculus, spectrum, complex analytic functions, Banach algebra, complex, generated by, associates, algebra, identity element, continuous, polynomial functions, contain, homomorphism, topological algebra, unital, limit, series, partial sums, exponential function, matrix, polynomial, functions, scalar, expression, identity operator, linear operator, field, normed vector space
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This is version 9 of functional calculus, born on 2007-08-22, modified 2007-10-14.
Object id is 9882, canonical name is FunctionalCalculus.
Accessed 841 times total.

Classification:

AMS MSC:	46H30 (Functional analysis :: Topological algebras, normed rings and algebras, Banach algebras :: Functional calculus in topological algebras)
	47A60 (Operator theory :: General theory of linear operators :: Functional calculus)

Pending Errata and Addenda

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