In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. 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 not using f on its original function domain any longer. This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of f(x) = x² and M an n×n matrix. The idea of a functional calculus is to create a principled approach to this kind of overloading of the notation.

The easy case is to apply polynomial functions to a square matrix, extending what has just been discussed. The interesting examples move to operators on infinite-dimensional vector spaces, and functions 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.

For technical accounts see

• holomorphic functional calculus
• continuous functional calculus
• Borel functional calculus.