Английская Википедия:Functional calculus
Шаблон:Short description 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 (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative.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 <math> f </math> is a function, say a numerical function of a real number, and <math> M </math> is an operator, there is no particular reason why the expression <math> f(M) </math> should make sense. If it does, then we are no longer using <math> f </math> 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 <math> f(x) = x^2 </math> and <math> M </math> an <math> n\times n </math> 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 <math> T </math>. This family is an ideal in the ring of polynomials. Furthermore, it is a nontrivial ideal: let <math> n </math> be the finite dimension of the algebra of matrices, then <math> \{I, T, T^2, \ldots, T^n \} </math> is linearly dependent. So <math> \sum_{i=0}^n \alpha_i T^i = 0 </math> for some scalars <math> \alpha_i </math>, not all equal to 0. This implies that the polynomial <math> \sum_{i=0}^n \alpha_i x^i </math> lies in the ideal. Since the ring of polynomials is a principal ideal domain, this ideal is generated by some polynomial <math> m </math>. Multiplying by a unit if necessary, we can choose <math> m </math> to be monic. When this is done, the polynomial <math> m </math> is precisely the minimal polynomial of <math> T </math>. This polynomial gives deep information about <math> T </math>. For instance, a scalar <math> \alpha </math> is an eigenvalue of <math> T </math> if and only if <math> \alpha </math> is a root of <math> m </math>. Also, sometimes <math> m </math> can be used to calculate the exponential of <math> T </math> 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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Шаблон:Analysis in topological vector spaces Шаблон:Spectral theory Шаблон:Functional analysis
