functional calculus
1 Basic Idea
Let $X$ be a normed vector space^{} over a field $\mathrm{\pi \x9d\x95\x82}$. 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
$$f\beta \x81\u2019(T)$$ 
to make sense as a linear operator in $X$, for certain scalar functions $f:\mathrm{\pi \x9d\x95\x82}\beta \x9f\u0386\mathrm{\pi \x9d\x95\x82}$.
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^{} $f\beta \x81\u2019(x)={a}_{k}\beta \x81\u2019{x}^{k}+\mathrm{\beta \x80\xa6}+{a}_{2}\beta \x81\u2019{x}^{2}+{a}_{1}\beta \x81\u2019x+{a}_{0}$, the expression
$$f\beta \x81\u2019(T):={a}_{k}\beta \x81\u2019{T}^{k}+\mathrm{\beta \x80\xa6}+{a}_{2}\beta \x81\u2019{T}^{2}+{a}_{1}\beta \x81\u2019T+{a}_{0}\beta \x81\u2019I$$ 
does indeed refer to a linear operator in $X$.
As another example, when $T$ is a matrix in ${\mathrm{\beta \x84\x9d}}^{n}$ or ${\mathrm{\beta \x84\x82}}^{n}$ one is sometimes led to the exponential^{} (http://planetmath.org/MatrixExponential) of $T$
$${e}^{T}=\underset{k=0}{\overset{\mathrm{\beta \x88\x9e}}{\beta \x88\x91}}\frac{{T}^{k}}{k!}$$ 
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 $f\beta \x81\u2019(T)$ if $f$ can be approximated by polynomials:
If $f$ can be approximated by polynomials ${p}_{n}$ then one could try to define
$$f\beta \x81\u2019(T):=\underset{n\beta \x86\x92\mathrm{\beta \x88\x9e}}{lim}\beta \x81\u2018{p}_{n}\beta \x81\u2019(T)$$ 
But for that one needs to define what βapproximatedβ 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^{} $\mathrm{\pi \x9d\x92\x9c}$ over a field $\mathrm{\pi \x9d\x95\x82}$.
There is no definition in mathematics of functional calculus, but the ideas above show that a functional calculus for an element $T\beta \x88\x88\mathrm{\pi \x9d\x92\x9c}$ should be something like an homomorphism^{} $(\beta \x8b\x85)\beta \x81\u2019(T):\mathrm{\beta \x84\pm}\beta \x9f\u0386\mathrm{\pi \x9d\x92\x9c}$ from some topological algebra of scalar functions $\mathrm{\beta \x84\pm}$ to $\mathrm{\pi \x9d\x92\x9c}$, that satisfied the following :

β’
$\mathrm{\beta \x84\pm}$ must contain the polynomial functions.

β’
$(\beta \x8b\x85)\beta \x81\u2019(T)$ is continuous^{}.

β’
$p\beta \x81\u2019(T)={a}_{k}\beta \x81\u2019{T}^{k}+\mathrm{\beta \x80\xa6}+{a}_{2}\beta \x81\u2019{T}^{2}+{a}_{1}\beta \x81\u2019T+{a}_{0}\beta \x81\u2019I$ for each polynomial $p\beta \x81\u2019(x)={a}_{k}\beta \x81\u2019{x}^{k}+\mathrm{\beta \x80\xa6}+{a}_{2}\beta \x81\u2019{x}^{2}+{a}_{1}\beta \x81\u2019x+{a}_{0}$, where $I$ denotes the identity element^{} of $\mathrm{\pi \x9d\x92\x9c}$.
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 $\mathrm{\pi \x9d\x92\x9c}$. 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^{} $\mathrm{\pi \x9d\x92\x9c}$. 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^{} $\mathrm{\pi \x9d\x92\x9c}$. 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 $f\beta \x81\u2019(\mathrm{{\rm O}\x83}\beta \x81\u2019(T))=\mathrm{{\rm O}\x83}\beta \x81\u2019(f\beta \x81\u2019(T))$, where $\mathrm{{\rm O}\x83}\beta \x81\u2019(\beta \x8b\x85)$ 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 

Canonical name  FunctionalCalculus 
Date of creation  20130322 17:29:40 
Last modified on  20130322 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 