# 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 (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative.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 $\displaystyle{ f }$ is a function, say a numerical function of a real number, and $\displaystyle{ M }$ is an operator, there is no particular reason why the expression $\displaystyle{ f(M) }$ should make sense. If it does, then we are no longer using $\displaystyle{ f }$ 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 $\displaystyle{ f(x) = x^2 }$ and $\displaystyle{ M }$ an $\displaystyle{ n\times n }$ 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 $\displaystyle{ T }$. This family is an ideal in the ring of polynomials. Furthermore, it is a nontrivial ideal: let $\displaystyle{ n }$ be the finite dimension of the algebra of matrices, then $\displaystyle{ \{I, T, T^2, \ldots, T^n \} }$ is linearly dependent. So $\displaystyle{ \sum_{i=0}^n \alpha_i T^i = 0 }$ for some scalars $\displaystyle{ \alpha_i }$, not all equal to 0. This implies that the polynomial $\displaystyle{ \sum_{i=0}^n \alpha_i x^i }$ lies in the ideal. Since the ring of polynomials is a principal ideal domain, this ideal is generated by some polynomial $\displaystyle{ m }$. Multiplying by a unit if necessary, we can choose $\displaystyle{ m }$ to be monic. When this is done, the polynomial $\displaystyle{ m }$ is precisely the minimal polynomial of $\displaystyle{ T }$. This polynomial gives deep information about $\displaystyle{ T }$. For instance, a scalar $\displaystyle{ \alpha }$ is an eigenvalue of $\displaystyle{ T }$ if and only if $\displaystyle{ \alpha }$ is a root of $\displaystyle{ m }$. Also, sometimes $\displaystyle{ m }$ can be used to calculate the exponential of $\displaystyle{ T }$ 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.