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In mathematics, an operator is generally a mapping or function that acts on elements of a space to produce elements of another space (possibly the same space, sometimes required to be the same space). There is no general definition of an operator, but the term is often used in place of function when the domain is a set of functions or other structured objects. Also, the domain of an operator is often difficult to be explicitly characterized (for example in the case of an integral operator), and may be extended to related objects (an operator that acts on functions may act also on differential equations whose functions are solutions). See Operator (physics) for other examples.

The most basic operators (in some sense) are linear maps, which act on vector spaces. However, when using "linear operator" instead of "linear map", mathematicians often mean actions on vector spaces of functions, which also preserve other properties, such as continuity. For example, differentiation and indefinite integration are linear operators; operators that are built from them are called differential operators, integral operators or integro-differential operators.

Operator is also used for denoting the symbol of a mathematical operation. This is related with the meaning of "operator" in computer programming, see operator (computer programming).

## Linear operators

The most common kind of operator encountered are linear operators. Let U and V be vector spaces over a field K. A mapping A: UV is linear if

${\displaystyle A(\alpha \mathbf {x} +\beta \mathbf {y} )=\alpha A\mathbf {x} +\beta A\mathbf {y} }$

for all x, y in U and for all α, β in K. This means that a linear operator preserves vector space operations, in the sense that it does not matter whether you apply the linear operator before or after the operations of addition and scalar multiplication. In more technical words, linear operators are morphi

The most basic operators (in some sense) are linear maps, which act on vector spaces. However, when using "linear operator" instead of "linear map", mathematicians often mean actions on vector spaces of functions, which also preserve other properties, such as continuity. For example, differentiation and indefinite integration are linear operators; operators that are built from them are called differential operators, integral operators or integro-differential operators.

Operator is also used for denoting the symbol of a mathematical operation. This is related with the meaning of "operator" in computer programming, see operator (computer programming).

The most common kind of operator encountered are linear operators. Let U and V be vector spaces over a field K. A mapping A: UV is linear if

${\displaystyle A(\alpha \mathbf {x} +\beta \mathbf {y} )=\alpha A\mathbf {x} +\beta A\mathbf {y} }$

for all x, y in U and for all α, β in K. This means that a linear operator preserves vector space operations, in the sense that it does not matter whether you apply the linear operator before or after the operations of addition and scalar multiplication. In more technical words, linear operators are morphisms between vector spaces.

In the finite-dimensional case linear operators can be represented by matrices in the following way. Let ${\displaystyle K$

for all x, y in U and for all α, β in K. This means that a linear operator preserves vector space operations, in the sense that it does not matter whether you apply the linear operator before or after the operations of addition and scalar multiplication. In more technical words, linear operators are morphisms between vector spaces.

In the finite-dimensional case linear operators can be represented by matrices in the following way. Let ${\displaystyle K}$ be a field, and ${\displaystyle U}$ and matrices in the following way. Let ${\displaystyle K}$ be a field, and ${\displaystyle U}$ and ${\displaystyle V}$ be finite-dimensional vector spaces over ${\displaystyle K}$. Let us select a basis ${\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}}$ in ${\displaystyle U}$ and ${\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{m}}$ in ${\displaystyle V}$. Then let ${\displaystyle \mathbf {x} =x^{i}\mathbf {u} _{i}}$ be an arbitrary vector in ${\displaystyle U}$ (assuming Einstein convention), and ${\displaystyle A:U\to V}$ be a linear operator. Then

Then ${\displaystyle a_{i}^{j}:=(A\mathbf {u} _{i})^{j}\in K}$ is the matrix of the operator ${\displaystyle A}$ in fixed bases. ${\displaystyle a_{i}^{j}}$ does not depend on the choice of ${\displaystyle x}$, and ${\displaystyle A\mathbf {x} =\mathbf {y} }$ if ${\displaystyle a_{i}^{j}x^{i}=y^{j}}$. Thus in fixed bases n-by-m matrices are in bijective correspondence to linear operators from ${\displaystyle U}$ to ${\displaystyle V}$.

The important concepts directly related to operators between finite-dimensional vector spaces are the ones of rank, determinant, inverse operator, and eigenspace.

Linear operators also play a great role in the infinite-dimensional case. The concepts of rank and determinant cannot be extended to infinite-dimensional matrices. This is why very different techniques are employed when studying linear operators (and operators in general) in the infinite-dimensional case. The study of linear operators in the infinite-dimensional case is known as functional analysis (so called because various classes of functions form interesting examples of infinite-dimensional vector spaces).

The space of sequences of real numbers, or more generally sequences of vectors in any vector space, themselves form an infinite-dimensional vector space. The most important cases are sequences of real or complex numbers, and these spaces, together with linear subspaces, are known as sequence spaces. Operators on these spaces are known as sequence transformations.

Bounded linear operators over Banach space form a Banach algebra in respect to the standard operator norm. The theory of Banach algebras develops a very general concept of spectra that elegantly generalizes the theory of eigenspaces.