# Functional calculus

In mathematics, a **functional calculus** is a theory allowing one to apply mathematical functions to mathematical operators. 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 (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative.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 *f* is a function, say a numerical function of a real number, and *M* is an operator, there is no particular reason why the expression

*f*(*M*)

should make sense. If it does, then we are no longer using *f* 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 *f*(*x*) = *x*^{2} and *M* an *n*×*n* matrix. The idea of a functional calculus is to create a *principled* approach to this kind of overloading of the notation.

The most immediate case is to apply polynomial functions 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 *T*. This family is an ideal in the ring of polynomials. Furthermore, it is a nontrivial ideal: let *n* be the finite dimension of the algebra of matrices, then {*I*, *T*, *T*^{2}...*T ^{n}*} is linearly dependent. So ∑

*α*= 0 for some scalars

_{i}T^{i}*α*, not all equal to 0. This implies that the polynomial ∑

_{i}*α*lies in the ideal. Since the ring of polynomials is a principal ideal domain, this ideal is generated by some polynomial

_{i}x^{i}*m*. Multiplying by a unit if necessary, we can choose

*m*to be monic. When this is done, the polynomial

*m*is precisely the minimal polynomial of

*T*. This polynomial gives deep information about

*T*. For instance, a scalar

*α*is an eigenvalue of

*T*if and only if

*α*is a root of

*m*. Also, sometimes

*m*can be used to calculate the exponential of

*T*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.

## References

- Hazewinkel, Michiel, ed. (2001) [1994], "Functional calculus",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4