functional calculus

1 Basic Idea

Let $X$ be a normed vector space over a field $\mathbb{K}$ . 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 $f:\mathbb{K}\longrightarrow\mathbb{K}$ .

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 $f(x)=a_{k}x^{k}+\dots+a_{2}x^{2}+a_{1}x+a_{0}$ , the expression

f(T):=a_{k}T^{k}+\dots+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 $\mathbb{R}^{n}$ or $\mathbb{C}^{n}$ one is sometimes led to the exponential (http://planetmath.org/MatrixExponential) of $T$

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 $e^{T}$ 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\rightarrow\infty}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 $\mathcal{A}$ 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 :

•

$\mathcal{F}$ must contain the polynomial functions.
•

$(\cdot)(T)$ is continuous.
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$p(T)=a_{k}T^{k}+\dots+a_{2}T^{2}+a_{1}T+a_{0}I$ for each polynomial $p(x)=a_{k}x^{k}+\dots+a_{2}x^{2}+a_{1}x+a_{0}$ , where $I$ denotes the identity element of $\mathcal{A}$ .

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 $\mathcal{A}$ . It associates polynomials to elements in the algebra generated by $T$ , as discussed above.
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functional calculus -

This is valid for any element $T$ in a complex Banach algebra $\mathcal{A}$ . 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 $\mathcal{A}$ . 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

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Functional calculi provide an of constructing new linear operators having specified out of given ones.
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There are strong with spectral theory since one usually has $f(\sigma(T))=\sigma(f(T))$ , where $\sigma(\cdot)$ denotes the spectrum of its . This is called the spectral mapping theorem.
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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 $\sqrt{T}$ 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