# 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 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 βapproximatedβ 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 :

• β’

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

• β’

$(\cdot)(T)$ is continuous.

• β’

$p(T)=a_{k}T^{k}+\dots+a_{2}T^{2}+a_{1}T+a_{0}I$ 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).

• β’

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.

• β’

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 $T$.

• β’

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 $T$.

• β’

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 $T$.

• β’

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

• β’

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

• β’

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.

• β’

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 FunctionalCalculus 2013-03-22 17:29:40 2013-03-22 17:29:40 asteroid (17536) asteroid (17536) 16 asteroid (17536) Feature msc 47A60 msc 46H30 FunctionalCalculusForHermitianMatrices ContinuousFunctionalCalculus2 PolynomialFunctionalCalculus BorelFunctionalCalculus