In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis.
If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory.
Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category.
Spectral theorem
The spectral theorem is any of a number of results about linear operators or about matrices.Sunder, V.S. Functional Analysis: Spectral Theory (1997) Birkhäuser Verlag In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This concept of diagonalization is relatively straightforward for operators on finite-dimensional spaces, but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modelled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.
Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces.
The spectral theorem also provides a canonical decomposition, called the spectral decomposition, eigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts.
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En son coeur, c'est un cours d'analyse fonctionnelle pour les physiciens et traite les bases de théorie de mesure, des espaces des fonctions et opérateurs linéaires.
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In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study of operator algebras are often phrased in algebraic terms, while the techniques used are often highly analytic. Although the study of operator algebras is usually classified as a branch of functional analysis, it has direct applications to representation theory, differential geometry, quantum statistical mechanics, quantum information, and quantum field theory.
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