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Operator theory
 
 
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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, operator theory is the branch of functional analysis
Functional analysis

Functional analysis is the branch of mathematics, and specifically of mathematical analysis, concerned with the study of vector spaces and operators acting upon them....
 which deals with bounded linear operators and their properties. It can be split crudely into two branches, although there is considerable overlap and interplay between them. These extend the 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. The name was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables....
 for bounded operators.








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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, operator theory is the branch of functional analysis
Functional analysis

Functional analysis is the branch of mathematics, and specifically of mathematical analysis, concerned with the study of vector spaces and operators acting upon them....
 which deals with bounded linear operators and their properties. It can be split crudely into two branches, although there is considerable overlap and interplay between them. These extend the 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. The name was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables....
 for bounded operators.

Single operator theory

Single operator theory deals with the properties and classification of single operators. For example, the classification of normal operator
Normal operator

In mathematics, especially functional analysis, a 'normal operator' on a complex Hilbert space is a continuous function linear operatorthat commutator with its hermitian adjoint N*:...
s in terms of their spectra falls into this category.

Operator algebras

The theory of operator algebra
Operator algebra

In functional analysis, an operator algebra is an algebra over a field of continuous function linear operators on a topological vector space with the multiplication given by the composition of mappings....
s brings algebra
Algebra over a field

In mathematics, an algebra over a field is an algebraic structure consisting of a vector space together with an Binary operation, usually called multiplication, that combines any two vectors to form a third vector....
s of operators such as C*-algebra
C*-algebra

C*-algebras are an important area of research in functional analysis, a branch of mathematics. The prototypical example of a C*-algebra is a complex number algebra over a field A of linear operators on a complex number Hilbert space with two additional properties:...
s to the fore.

See also

  • Fredholm theory
    Fredholm theory

    In mathematics, Fredholm theory is a theory of integral equations. In the narrowest sense, Fredholm theory concerns itself with the solution of the Fredholm integral equation....


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