functional calculus

{{Short description|Theory allowing one to apply mathematical functions to mathematical operators}}

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 (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative.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 $n\backslash times\; n$ 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 $\backslash \{I,\; T,\; T^2,\; \backslash ldots,\; T^N\; \backslash \}$ is linearly dependent. So $\backslash sum\_\{i=0\}^N\; \backslash alpha\_i\; T^i\; =\; 0$ for some scalars $\backslash alpha\_i$, not all equal to 0. This implies that the polynomial $\backslash sum\_\{i=0\}^N\; \backslash alpha\_i\; x^i$ 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 $\backslash alpha$ is an eigenvalue of $T$ if and only if $\backslash alpha$ 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.