View source for Functional calculus

{{Short description|Theory allowing one to apply mathematical functions to mathematical operators}}
{{no footnotes|date=September 2025}}
In [[mathematics]], a '''functional calculus''' is a theory allowing one to apply [[mathematical function]]s to [[mathematical operator]]s.<ref>https://users.metu.edu.tr/baver/chapter5.pdf

from Preface on page 8.</ref> It is now a branch (more accurately, several related areas) of the field of [[functional analysis]], connected with [[spectral theory]]. (Historically, the term was also used synonymously with [[calculus of variations]]; this usage is obsolete, except for [[functional derivative]]. Sometimes it is used in relation to types of [[functional equations]], or in logic for systems of [[predicate calculus]].)

If <math> f </math> is a function, say a numerical function of a [[real number]], and <math> M </math> is an operator, there is no particular reason why the expression <math> f(M) </math> should make sense. If it does, then we are no longer using <math> f </math> on its original [[function domain]]. In the tradition of [[operational calculus]], algebraic expressions in operators are handled irrespective of their meaning.  This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of <math> f(x) = x^2 </math> and <math> M </math> an <math> n\times n </math> [[matrix (mathematics)|matrix]]. The idea of a functional calculus is to create a ''principled'' approach to this kind of [[Function overloading|overloading]] of the notation.

The most immediate case is to apply [[polynomial function]]s to a [[square matrix]], extending what has just been discussed. In the finite-dimensional case, the polynomial functional calculus yields quite a bit of information about the operator. For example, consider the family of polynomials which annihilates an operator <math> T </math>. This family is an [[ideal (ring theory)|ideal]] in the ring of polynomials. Furthermore, it is a nontrivial ideal: let <math> N </math> be the finite dimension of the algebra of matrices, then <math> \{I, T, T^2, \ldots, T^N \} </math> is linearly dependent. So <math> \sum_{i=0}^N \alpha_i T^i = 0 </math> for some scalars <math> \alpha_i </math>, not all equal to 0. This implies that the polynomial <math> \sum_{i=0}^N \alpha_i x^i </math> lies in the ideal. Since the ring of polynomials is a [[principal ideal domain]], this ideal is generated by some polynomial <math> m </math>. Multiplying by a unit if necessary, we can choose <math> m </math> to be monic. When this is done, the polynomial <math> m </math> is precisely the [[Minimal polynomial (linear algebra)|minimal polynomial]] of <math> T </math>. This polynomial gives deep information about <math> T </math>. For instance, a scalar <math> \alpha </math> is an eigenvalue of <math> T </math> if and only if <math> \alpha </math> is a root of <math> m </math>. Also, sometimes <math> m </math> can be used to calculate the [[Exponential function|exponential]] of <math> T </math> efficiently.

The polynomial calculus is not as informative in the infinite-dimensional case. Consider the [[unilateral shift]] with the polynomials calculus; the ideal defined above is now trivial. Thus one is interested in functional calculi more general than polynomials. The subject is closely linked to [[spectral theory]], since for a [[diagonal matrix]] or [[multiplication operator]], it is rather clear what the definitions should be.

==See also==

* {{annotated link|Borel functional calculus}}
* {{annotated link|Continuous functional calculus}}
* {{annotated link|Direct integral}}
* {{annotated link|Holomorphic functional calculus}}

==References==

{{reflist}}

== Further reading ==
* {{Springer|id=F/f042030|title=Functional calculus}}

==External links==
* {{Commons category-inline}}

{{Analysis in topological vector spaces}}
{{Spectral theory}}
{{Functional analysis}}

{{DEFAULTSORT:Functional Calculus}}
[[Category:Functional calculus| ]]