Operator (mathematics)

The most common kind of operators encountered are linear operators. Let U and V be vector spaces over some field K . A mapping ${\displaystyle \ \operatorname {A} :U\to V\ }$ is linear if

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

for all x in the vector space U and y in the vector space V, and for all α, β in their associated field 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 K be a field, and ${\displaystyle U}$ and V be finite-dimensional vector spaces over K . Let us select a basis ${\displaystyle \ \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}\ }$ in U and ${\displaystyle \ \mathbf {v} _{1},\ldots ,\mathbf {v} _{m}\ }$ in V . Then let ${\displaystyle \ \mathbf {x} =x^{i}\mathbf {u} _{i}}$ be an arbitrary vector in ${\displaystyle U}$ (assuming Einstein convention), and ${\displaystyle \ \operatorname {A} :U\to V\ }$ be a linear operator. Then

$${\displaystyle \ \operatorname {A} \mathbf {x} =x^{i}\operatorname {A} \mathbf {u} _{i}=x^{i}\left(\operatorname {A} \mathbf {u} _{i}\right)^{j}\mathbf {v} _{j}~.}$$

Then ${\displaystyle \ a_{i}^{j}\equiv \left(\operatorname {A} \mathbf {u} _{i}\right)^{j}\ ,}$ with all ${\displaystyle \ a_{i}^{j}\in K\ ,}$ is the matrix form of the operator ${\displaystyle \operatorname {A} }$ in the fixed basis ${\displaystyle \ \{\mathbf {u} _{i}\}_{i=1}^{n}~.}$ The tensor ${\displaystyle \ a_{i}^{j}\ }$ does not depend on the choice of ${\displaystyle \ x\ ,}$ and ${\displaystyle \;\operatorname {A} \mathbf {x} =\mathbf {y} \;}$ 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.

Let U and V be two vector spaces over the same ordered field (for example, ${\displaystyle \ \mathbb {R} \ }$), and they are equipped with norms. Then a linear operator from U to V is called bounded if there exists c > 0 such that

$${\displaystyle \ \|\operatorname {A} \mathbf {x} \|_{V}\leq c\ \|\mathbf {x} \|_{U}\ }$$

for every x in U . Bounded operators form a vector space. On this vector space we can introduce a norm that is compatible with the norms of U and V:

$${\displaystyle \ \|\operatorname {A} \|=\inf\{\ c:\|\operatorname {A} \mathbf {x} \|_{V}\leq c\ \|\mathbf {x} \|_{U}\}~.}$$

In case of operators from U to itself it can be shown that

${\textstyle \ \|\operatorname {A} \operatorname {B} \|\leq \|\operatorname {A} \|\cdot \|\operatorname {B} \|~.}$ [lower-alpha 2]

Any unital normed algebra with this property is called a Banach algebra. It is possible to generalize spectral theory to such algebras. C*-algebras, which are Banach algebras with some additional structure, play an important role in quantum mechanics.

• Function
• Operator algebra
• List of operators