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 (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative.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 $$ n\times n $$ 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 $$ T $$. 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 $$ \{I, T, T^2, \ldots, T^n \} $$ is linearly dependent. So $$ \sum_{i=0}^n \alpha_i T^i = 0 $$ for some scalars $$ \alpha_i $$, not all equal to 0. This implies that the polynomial $$ \sum_{i=0}^n \alpha_i x^i $$ lies in the ideal. Since the ring of polynomials is a principal ideal domain, this ideal is generated by some polynomial $$ m $$. 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 $$ \alpha $$ is an eigenvalue of $$ T $$ if and only if $$ \alpha $$ 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.