# functional calculus

## 1 Basic Idea

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 $T$

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

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 $\mathbb{K}$.

## 4 Applications

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