Functional calculus (original) (raw)

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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 (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative.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 {\displaystyle f} {\displaystyle f} is a function, say a numerical function of a real number, and M {\displaystyle M} {\displaystyle M} is an operator, there is no particular reason why the expression f ( M ) {\displaystyle f(M)} {\displaystyle f(M)} should make sense. If it does, then we are no longer using f {\displaystyle f} {\displaystyle 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 {\displaystyle f(x)=x^{2}} {\displaystyle f(x)=x^{2}} and M {\displaystyle M} {\displaystyle M} an n × n {\displaystyle n\times n} {\displaystyle 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 {\displaystyle T} {\displaystyle T}. This family is an ideal in the ring of polynomials. Furthermore, it is a nontrivial ideal: let n {\displaystyle n} {\displaystyle n} be the finite dimension of the algebra of matrices, then { I , T , T 2 , … , T n } {\displaystyle \{I,T,T^{2},\ldots ,T^{n}\}} {\displaystyle \{I,T,T^{2},\ldots ,T^{n}\}} is linearly dependent. So ∑ i = 0 n α i T i = 0 {\displaystyle \sum _{i=0}^{n}\alpha _{i}T^{i}=0} {\displaystyle \sum _{i=0}^{n}\alpha _{i}T^{i}=0} for some scalars α i {\displaystyle \alpha _{i}} {\displaystyle \alpha _{i}}, not all equal to 0. This implies that the polynomial ∑ i = 0 n α i x i {\displaystyle \sum _{i=0}^{n}\alpha _{i}x^{i}} {\displaystyle \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 {\displaystyle m} {\displaystyle m}. Multiplying by a unit if necessary, we can choose m {\displaystyle m} {\displaystyle m} to be monic. When this is done, the polynomial m {\displaystyle m} {\displaystyle m} is precisely the minimal polynomial of T {\displaystyle T} {\displaystyle T}. This polynomial gives deep information about T {\displaystyle T} {\displaystyle T}. For instance, a scalar α {\displaystyle \alpha } {\displaystyle \alpha } is an eigenvalue of T {\displaystyle T} {\displaystyle T} if and only if α {\displaystyle \alpha } {\displaystyle \alpha } is a root of m {\displaystyle m} {\displaystyle m}. Also, sometimes m {\displaystyle m} {\displaystyle m} can be used to calculate the exponential of T {\displaystyle T} {\displaystyle 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.