Functional calculus explained

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 (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative.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 x 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 ^{n} _{i =0} \alpha _{i} T ^{i} = 0 for some scalars \alpha _{i} , not all equal to 0. This implies that the polynomial \sum ^{n} _{i =0} \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.