functional calculus

1 Basic Idea

Let X be a normed vector spacePlanetmathPlanetmath over a field 𝕂. Let T be a linear operatorMathworldPlanetmath in X and I the identity operatorMathworldPlanetmath in X.

The functional calculusMathworldPlanetmath refers to a specific process which enables the expression


to make sense as a linear operator in X, for certain scalar functions f:π•‚βŸΆπ•‚.

At first sight, and for most functionsMathworldPlanetmath f, there is no reason why the above expression should be associated with a particular linear operator.

But, for example, when f is a polynomialPlanetmathPlanetmath f⁒(x)=ak⁒xk+…+a2⁒x2+a1⁒x+a0, the expression


does indeed refer to a linear operator in X.

As another example, when T is a matrix in ℝn or β„‚n one is sometimes led to the exponentialMathworldPlanetmathPlanetmath ( of T


Thus, we are applying the scalar exponential functionDlmfDlmfMathworld to a matrix.

Note in this last example that eT 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 pn then one could try to define


But for that one needs to define what β€œapproximated” 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 algebraMathworldPlanetmath π’œ over a field 𝕂.

There is no definition in mathematics of functional calculus, but the ideas above show that a functional calculus for an element Tβˆˆπ’œ should be something like an homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath (β‹…)⁒(T):β„±βŸΆπ’œ from some topological algebra of scalar functions β„± to π’œ, that satisfied the following :

  • β€’

    β„± must contain the polynomial functions.

  • β€’

    (β‹…)⁒(T) is continuousMathworldPlanetmathPlanetmath.

  • β€’

    p⁒(T)=ak⁒Tk+…+a2⁒T2+a1⁒T+a0⁒I for each polynomial p⁒(x)=ak⁒xk+…+a2⁒x2+a1⁒x+a0, where I denotes the identity elementMathworldPlanetmath of π’œ.

3 Functional Calculi

There are some functionalMathworldPlanetmathPlanetmathPlanetmath calculi of . We give a very brief descprition of each one of them (please follows the links for entries with more detailed explanation).

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⁒(σ⁒(T))=σ⁒(f⁒(T)), where σ⁒(β‹…) 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 rootMathworldPlanetmath T of an T.

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