# polynomial functional calculus

Let $\mathcal{A}$ be an unital associative algebra over $\u2102$ with identity element^{} $e$ and let $a\in \mathcal{A}$.

The polynomial functional calculus is the most basic form of a functional calculus. It allows the expression

$p(a)$ |

to make sense as an element of $\mathcal{A}$, for any polynomial^{} $p:\u2102\u27f6\u2102$.

This is achieved in the following natural way: for any polynomial $p(\lambda ):=\sum {c}_{n}{\lambda}^{n}$ we the element $p(a):=\sum {c}_{n}{a}^{n}\in \mathcal{A}$.

## 1 Definition

Recall that the set of polynomial functions in $\u2102$, denoted by $\u2102[\lambda ]$, is an associative algebra over $\u2102$ under pointwise operations and is generated by the constant polynomial $1$ and the variable^{} $\lambda $ (corresponding to the identity function in $\u2102$).

Moreover, any homomorphism^{} from the algebra^{} $\u2102[\lambda ]$ is perfectly determined by the values of $1$ and $\lambda $.

Definition - *Consider the algebra homomorphism $\pi \mathrm{:}\mathrm{C}\mathit{}\mathrm{[}\lambda \mathrm{]}\mathrm{\u27f6}\mathrm{A}$ such that $\pi \mathit{}\mathrm{(}\mathrm{1}\mathrm{)}\mathrm{=}e$ and $\pi \mathit{}\mathrm{(}\lambda \mathrm{)}\mathrm{=}a$. This homomorphism is denoted by*

$p\u27fcp(a)$ |

*and it is called the* polynomial functional calculus *for $a$.*

It is clear that for any polynomial $p(\lambda ):=\sum {c}_{n}{\lambda}^{n}$ we have $p(a)=\sum {c}_{n}{a}^{n}$.

## 2 Spectral Properties

We will denote by $\sigma (x)$ the spectrum (http://planetmath.org/Spectrum) of an element $x\in \mathcal{A}$.

Theorem^{} - (polynomial spectral mapping theorem) - *Let $\mathrm{A}$ be an unital associative algebra over $\mathrm{C}$ and $a$ an element in $\mathrm{A}$. For any polynomial $p$ we have that*

$\sigma (p(a))=p(\sigma (a))$ |

*:* Let us first prove that $\sigma (p(a))\subseteq p(\sigma (a))$. Suppose $\stackrel{~}{\lambda}\in \sigma (p(a))$, which means that $p(a)-\stackrel{~}{\lambda}e$ is not invertible^{}. Now consider the polynomial in $\u2102$ given by $q:=p-\stackrel{~}{\lambda}$. It is clear that $q(a)=p(a)-\stackrel{~}{\lambda}e$, and therefore $q(a)$ is not invertible. Since $\u2102$ is algebraically closed^{} (http://planetmath.org/FundamentalTheoremOfAlgebra), we have that

$q(\lambda )={(\lambda -{\lambda}_{1})}^{{n}_{1}}\mathrm{\cdots}{(\lambda -{\lambda}_{k})}^{{n}_{k}}$ |

for some ${\lambda}_{1},\mathrm{\dots},{\lambda}_{k}\in \u2102$ and ${n}_{1},\mathrm{\dots},{n}_{k}\in \mathbb{N}$. Thus, we can also write a similar product^{} for $q(a)$ as

$q(a)={(a-{\lambda}_{1}e)}^{{n}_{1}}\mathrm{\cdots}{(a-{\lambda}_{k}e)}^{{n}_{k}}$ |

Now, since $q(a)$ is not invertible we must have that at least one of the factors $(a-{\lambda}_{i}e)$ is not invertible, which means that for that particular ${\lambda}_{i}$ we have ${\lambda}_{i}\in \sigma (a)$. But we also have that $q({\lambda}_{i})=0$, i.e. $p({\lambda}_{i})=\stackrel{~}{\lambda}$, and hence $\stackrel{~}{\lambda}\in p(\sigma (a))$.

We now prove the inclusion $\sigma (p(a))\supseteq p(\sigma (a))$. Suppose $\stackrel{~}{\lambda}\in p(\sigma (a))$, which means that $\stackrel{~}{\lambda}=p({\lambda}_{0})$ for some ${\lambda}_{0}\in \sigma (a)$. The polynomial $p-\stackrel{~}{\lambda}$ has a zero at ${\lambda}_{0}$, hence there is a polynomial $d$ such that

$p(\lambda )-\stackrel{~}{\lambda}=d(\lambda )(\lambda -{\lambda}_{0}),\lambda \in \u2102$ |

Thus, we can also write a similar product for $q(a)$ as

$p(a)-\stackrel{~}{\lambda}e=d(a)(a-{\lambda}_{0}e)$ |

If $p(a)-\stackrel{~}{\lambda}e$ was invertible, then we would see that $a-{\lambda}_{0}e$ had a left (http://planetmath.org/InversesInRings) and a right inverse^{} (http://planetmath.org/InversesInRings), thus being invertible. But we know that ${\lambda}_{0}\in \sigma (a)$, hence we conclude that $p(a)-\stackrel{~}{\lambda}e$ cannot be invertible, i.e. $\stackrel{~}{\lambda}\in \sigma (p(a))$. $\mathrm{\square}$

Title | polynomial functional calculus |
---|---|

Canonical name | PolynomialFunctionalCalculus |

Date of creation | 2013-03-22 18:48:23 |

Last modified on | 2013-03-22 18:48:23 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 8 |

Author | asteroid (17536) |

Entry type | Feature |

Classification | msc 46H30 |

Classification | msc 47A60 |

Related topic | FunctionalCalculus |

Related topic | ContinuousFunctionalCalculus2 |

Related topic | BorelFunctionalCalculus |

Defines | polynomial spectral mapping theorem |