polynomial functional calculus
Let be an unital associative algebra over with identity element and let .
The polynomial functional calculus is the most basic form of a functional calculus. It allows the expression
to make sense as an element of , for any polynomial .
This is achieved in the following natural way: for any polynomial we the element .
1 Definition
Recall that the set of polynomial functions in , denoted by , is an associative algebra over under pointwise operations and is generated by the constant polynomial and the variable (corresponding to the identity function in ).
Moreover, any homomorphism from the algebra is perfectly determined by the values of and .
Definition - Consider the algebra homomorphism such that and . This homomorphism is denoted by
and it is called the polynomial functional calculus for .
It is clear that for any polynomial we have .
2 Spectral Properties
We will denote by the spectrum (http://planetmath.org/Spectrum) of an element .
Theorem - (polynomial spectral mapping theorem) - Let be an unital associative algebra over and an element in . For any polynomial we have that
: Let us first prove that . Suppose , which means that is not invertible. Now consider the polynomial in given by . It is clear that , and therefore is not invertible. Since is algebraically closed (http://planetmath.org/FundamentalTheoremOfAlgebra), we have that
for some and . Thus, we can also write a similar product for as
Now, since is not invertible we must have that at least one of the factors is not invertible, which means that for that particular we have . But we also have that , i.e. , and hence .
We now prove the inclusion . Suppose , which means that for some . The polynomial has a zero at , hence there is a polynomial such that
Thus, we can also write a similar product for as
If was invertible, then we would see that had a left (http://planetmath.org/InversesInRings) and a right inverse (http://planetmath.org/InversesInRings), thus being invertible. But we know that , hence we conclude that cannot be invertible, i.e. .
Title | polynomial functional calculus |
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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 |