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 synonymous with the calculus of variations; the latter term remains in extensive use in physics and engineering texts, whereas functional calculus develops the subject further with more mathematically careful, formal, abstract and precise articulations. The older usage is still visible in the functional derivative, which is often called the variational derivative.

Usage

There are several unrelated uses of the term "functional calculus": it is sometimes applied to types of functional equations, and sometimes to systems of logic in predicate calculus.

Some of the areas of mathematics that fall under the term "functional calculus" include:

• The operational calculus, a technique for solving differential equations by converting them into polynomial equations. The central idea is to view integration and differentiation as operators in their own right; differential equations then appear to have the algebraic form of polynomials in these operators.

• Holomorphic functional calculus, which attempts to extend the techniques commonly used to study holomorphic functions is a matrix or linear operator is a polynomial, leading to the "polynomial functional calculus".

• Continuous functional calculus, which considers the application of continuous functions to the normal elements of a C*-algebra. The ability to do this is the primary distinguishing feature that singles out the C*-algebras as a subset of Banach algebras, which support only the holomorphic functional calculus. The C*-algebras were originally developed to explore and formalize the operator equations being developed for quantum mechanics, specifically for matrix mechanics. The defining characteristics of quantum mechanics included the idea that the operators acted on Hilbert spaces, that there is a norm that expresses the expectation value of an operator, and that complex conjugation is an involution, expressing the adjoint of an operator, thus forming a *-algebra.

• Borel functional calculus, which generalizes the continuous functional calculus by considering any Borel function applied to an operator from a commutative algebra. For example, the Borel functional calculus can rigorously define the "square root" of the (negative) Laplacian operator

Informal motivation

The techniques developed for the calculus of variations is a polynomial is a matrix or a linear operator. To give this precise meaning, one must tackle issues of convergence, compactness, boundedness and continuity.

In the finite-dimensional case, the polynomial functional calculus. This family is an ideal lies in the ideal. Since the ring of polynomials is a principal ideal domain is precisely the minimal polynomial can be used to calculate the exponential 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.

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