functional calculus
1 Basic Idea
Let be a normed vector space over a field . Let be a linear operator in and the identity operator in .
The functional calculus refers to a specific process which enables the expression
to make sense as a linear operator in , for certain scalar functions .
At first sight, and for most functions , there is no reason why the above expression should be associated with a particular linear operator.
As another example, when is a matrix in or one is sometimes led to the exponential (http://planetmath.org/MatrixExponential) of
Thus, we are applying the scalar exponential function to a matrix.
Note in this last example that is approximated by polynomials (the partial sums of the series). This provides an idea of how to make sense of if can be approximated by polynomials:
If can be approximated by polynomials then one could try to define
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 over a field .
There is no definition in mathematics of functional calculus, but the ideas above show that a functional calculus for an element should be something like an homomorphism from some topological algebra of scalar functions to , that satisfied the following :
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β’
must contain the polynomial functions.
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β’
is continuous.
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β’
for each polynomial , where denotes the identity element of .
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).
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β’
This is valid for any element in any algebra . It associates polynomials to elements in the algebra generated by , as discussed above.
-
β’
functional calculus -
This is valid for any element in a complex Banach algebra . It associates complex analytic functions defined on the spectrum (http://planetmath.org/Spectrum) of to elements in the algebra generated by .
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β’
This is valid for normal elements in -algebras (http://planetmath.org/CAlgebra). It associates continuous functions on the spectrum of to elements in the -algebra generated by .
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β’
This is valid for normal operators in a von Neumann algebra . It associates bounded Borel measurable functions on the spectrum of to elements in the von Neumann algebra generated by .
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β’
A case. It is valid for Hermitian matrices . It associates real valued functions on the spectrum of to elements in the algebra generated by .
4 Applications
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β’
Functional calculi provide an of constructing new linear operators having specified out of given ones.
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β’
There are strong with spectral theory since one usually has , where denotes the spectrum of its . This is called the spectral mapping theorem.
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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 of an .
Title | functional calculus |
---|---|
Canonical name | FunctionalCalculus |
Date of creation | 2013-03-22 17:29:40 |
Last modified on | 2013-03-22 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 |