Functional calculus
Encyclopedia
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 obsolescent, see though functional derivative
. Sometimes it is used in relation to types of functional equation
, 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
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) = 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 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 {I, T, T2...Tn} is linearly dependent. So ∑ αi Ti = 0 for some scalars αi, not all equal to 0. This implies that the polynomial ∑ αi xi lies in the ideal. Since the ring of polynomials is a principal ideal domain
, this ideal is generated by some polynomial m. The polynomial m is precisely the minimal polynomial of T. One has, for instance, a scalar α is an eigenvalue of T if and only if α 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.
For technical accounts see:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, connected with spectral theory
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...
. (Historically, the term was also used synonymously with calculus of variations
Calculus of variations
Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...
; this usage is obsolescent, see though functional derivative
Functional derivative
In mathematics and theoretical physics, the functional derivative is a generalization of the gradient. While the latter differentiates with respect to a vector with discrete components, the former differentiates with respect to a continuous function. Both of these can be viewed as extensions of...
. Sometimes it is used in relation to types of functional equation
Functional equation
In mathematics, a functional equation is any equation that specifies a function in implicit form.Often, the equation relates the value of a function at some point with its values at other points. For instance, properties of functions can be determined by considering the types of functional...
, or in logic for systems of predicate calculus.)
If f is a function, say a numerical function of a real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
, 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
Operational calculus
Operational calculus, also known as operational analysis, is a technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomial equation.-History:...
, 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) = x2 and M an n×n matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
. The idea of a functional calculus is to create a principled approach to this kind of overloading
Overloading
The terms overloading and overloaded may refer to:*Constructor and function/method overloading, in computer science, a type of polymorphism where different functions with the same name are invoked based on the data types of the parameters passed...
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
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
in the ring of polynomials. Furthermore, it is a nontrivial ideal: let n be the finite dimension of the algebra of matrices, then {I, T, T2...Tn} is linearly dependent. So ∑ αi Ti = 0 for some scalars αi, not all equal to 0. This implies that the polynomial ∑ αi xi lies in the ideal. Since the ring of polynomials is a principal ideal domain
Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
, this ideal is generated by some polynomial m. The polynomial m is precisely the minimal polynomial of T. One has, for instance, a scalar α is an eigenvalue of T if and only if α is a root of m. Also, sometimes m can be used to calculate the exponential
Exponential function
In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...
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
Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...
, since for a diagonal matrix
Diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
or multiplication operator
Multiplication operator
In operator theory, a multiplication operator is a linear operator T defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f...
, it is rather clear what the definitions should be.
For technical accounts see:
- holomorphic functional calculusHolomorphic functional calculusIn mathematics, holomorphic functional calculus is functional calculus with holomorphic functions. That is to say, given a holomorphic function ƒ of a complex argument z and an operator T, the aim is to construct an operatorf\,...
- continuous functional calculusContinuous functional calculusIn mathematics, the continuous functional calculus of operator theory and C*-algebra theory allows applications of continuous functions to normal elements of a C*-algebra. More precisely,Theorem...
- Borel functional calculusBorel functional calculusIn functional analysis, a branch of mathematics, the Borel functional calculus is a functional calculus , which has particularly broad scope. Thus for instance if T is an operator, applying the squaring function s → s2 to T yields the operator T2...
.