Operator

A mapping of one set into another, each of which has a certain structure (defined by algebraic operations, a topology, or by an order relation). The general definition of an operator coincides with the definition of a mapping or function. Let $X$ and $Y$ be two sets. A rule or correspondence which assigns a uniquely defined element $A(x)\in Y$ to every element $x$ of a subset $D\subset X$ is called an operator $A$ from $X$ into $Y$. $$\begin{equation} A:D\to Y, \qquad \text{where } D \subset X. \end{equation}$$ The term operator is mostly used in the case where $X$ and $Y$ are vector spaces. The expression $A(x)$ is often written as $Ax$.

Definitions and Notations[edit]

• The subset $D$ is called the domain of definition of the operator $A$ and is denoted by $\operatorname{Dom}(A)$; the set $\{A(x) : x\in D\}$ is called the domain of values of the operator $A$ (or its range) and is denoted by $\operatorname{R}(A)$.
• If $A$ is an operator from $X$ into $Y$ where $X=Y$, then $A$ is called an operator on $X$.
• If $\operatorname{Dom}(A)=X$, then $A$ is called an everywhere-defined operator.
• If $A_1$, $A_2$ are operators from $X_1$ into $Y_1$ and from $X_2$ into $Y_2$ with domains of definition $\operatorname{Dom}(A_1)$ and $\operatorname{Dom}(A_2)$, respectively, such that $\operatorname{Dom}(A_1)\subset\operatorname{Dom}(A_2)$ and $A_1x=A_2x$ for all $x\in\operatorname{Dom}(A_1)$, then if $X_1=X_2$, $Y_1=Y_2$, the operator $A_1$ is called a compression or restriction of the operator $A_2$, while $A_2$ is called an extension of $A_1$; if $X_1\subset X_2$, $A_2$ is called an extension of $A_1$ exceeding $X_1$.
• If $X$ and $Y$ are vector spaces, then in the set of all operators from $X$ into $Y$ it is possible to single out the class of linear operators (cf. Linear operator); the remaining operators from $X$ into $Y$ are called non-linear operators.
• If $X$ and $Y$ are topological vector spaces, then in the set of operators from $X$ into $Y$ the class of continuous operators (cf. Continuous operator) can be naturally singled out, so are the class of bounded linear operators $A$ (operators $A$ such that the image of any bounded set in $X$ is bounded in $Y$) and the class of compact linear operators (i.e. operators such that the image of any bounded set in $X$ is pre-compact in $Y$, cf. Compact operator).
• If $X$ and $Y$ are locally convex spaces, then it is natural to examine different topologies on $X$ and $Y$; an operator is said to be semi-continuous if it defines a continuous mapping from the space $X$ (with the initial topology) into the space $Y$ with the weak topology (the concept of semi-continuity is mainly used in the theory of non-linear operators); an operator is said to be strongly continuous if it is continuous as a mapping from $X$ with the boundedly weak topology into the space $Y$; an operator is called weakly continuous if it defines a continuous mapping from $X$ into $Y$ where $X$ and $Y$ have the weak topology. Compact operators are often called completely-continuous operators. Sometimes the term "competely-continuous operator" is used instead of "strongly-continuous operator" , or to denote an operator which maps any weakly-convergent sequence to a strongly-convergent one; if $X$ and $Y$ are reflexive Banach spaces, then these conditions are equivalent to the compactness of the operator. If an operator is strongly continuous, then it is weakly continuous.

Connection with Equations[edit]

Many equations in function spaces or abstract spaces can be expressed in the form $Ax=y$, where $y\in Y$, $x \in X$; $y$ is given, $x$ is unknown and $A$ is an operator from $X$ into $Y$. The assertion of the existence of a solution to this equation for any right-hand side $y\in Y$ is equivalent to the assertion that the range of the operator $A$ is the whole space $Y$; the assertion that the equation $Ax=y$ has a unique solution for any $y\in\operatorname{R}(A)$ means that $A$ is a one-to-one mapping from $\operatorname{Dom}(A)$ onto $\operatorname{R}(A)$.

Graph[edit]

The set $\Gamma(A)\subset X\times Y$ defined by the relation $$\begin{equation} \Gamma(A) = \{(x,Ax) : x\in \operatorname{Dom}(A)\} \end{equation}$$ is called the graph of the operator $A$. Let $X$ and $Y$ be topological vector spaces; an operator from $X$ into $Y$ is called a closed operator if its graph is closed. The concept of a closed operator is particularly useful in the case of linear operators with a dense domain of definition.

The concept of a graph allows one to generalize the concept of an operator: Any subset $A$ in $X\times Y$ is called a multi-valued operator from $X$ into $Y$; if $X$ and $Y$ are vector spaces, then a linear subspace in $X\times Y$ is called a multi-valued linear operator; the set $$\begin{equation} D(A) = \{x\in X : \text{ there exists an } y\in Y \text{ such that } (x, y)\in A \} \end{equation}$$ is called the domain of definition of the multi-valued operator.

If $X$ is a vector space over a field $\mathcal K$ and $Y = \mathcal K$, then an everywhere-defined operator from $X$ into $Y$ is called a functional on $X$.

If $ X $ and $ Y $ are locally convex spaces, then an operator $ A $ from $ X $ into $ Y $ with a dense domain of definition in $ X $ has an adjoint operator $ A ^{*} $ with a dense domain of definition in $ Y ^{*} $( with the weak topology) if, and only if, $ A $ is a closed operator.

Examples of operators.[edit]

1) The operator assigning the element $ 0 \in Y $ to any element $ x \in X $( the zero operator).

2) The operator mapping each element $ x \in X $ to the same element $ x \in X $( the identity operator on $ X $, written as $ \mathop{\rm id}\nolimits _{X} $ or $ 1 _{X} $).

3) Let $ X $ be a vector space of functions on a set $ M $, and let $ f $ be a function on $ M $; the operator on $ X $ with domain of definition

$$ D(A) = \{ {\phi \in X} : {f \phi \in X} \} $$

and acting according to the rule

$$ A \phi = f \phi $$

if $ \phi \in D(A) $, is called the operator of multiplication by a function; $ A $ is a linear operator.

4) Let $ X $ be a vector space of functions on a set $ M $, and let $ F $ be a mapping from the set $ M $ into itself; the operator on $ X $ with domain of definition

$$ D(A) = \{ {\phi \in X} : {\phi \circ F \in X} \} $$

and acting according to the rule

$$ A \phi = \phi \circ F $$

if $ \phi \in D(A) $, is a linear operator.

5) Let $ X,\ Y $ be vector spaces of real measurable functions on two measure spaces $ (M,\ \Sigma _{M} ,\ \mu ) $ and $ (N,\ \Sigma _{N} ,\ \nu ) $, respectively, and let $ K $ be a function on $ M \times N \times \mathbf R $, measurable with respect to the product measure $ \mu \times \nu \times \mu _{0} $, where $ \mu _{0} $ is Lebesgue measure on $ \mathbf R $, and continuous in $ t \in \mathbf R $ for any fixed $ m \in M $, $ n \in N $. The operator from $ X $ into $ Y $ with domain of definition $ D(A) = \{ {\phi \in X} : {f(x) = \int _{M} K (x,\ y,\ \phi (y)) \ dy} \} $, which exists for almost-all $ x \in N $ and $ f \in Y $, and acting according to the rule $ A \phi = f $ if $ \phi \in D(A) $, is called an integral operator; if

$$ K(x,\ y,\ z) = K(x,\ y)z, x \in M, y \in N, z \in \mathbf R , $$

then $ A $ is a linear operator.

6) Let $ X $ be a vector space of functions on a differentiable manifold $ M $, let $ \xi $ be a vector field on $ M $; the operator $ A $ on $ X $ with domain of definition

$$ D(A) = \{ {f \in X} : {\textrm{ the derivative } D _ \xi f \textrm{ of the function } f \textrm{ along the field } \xi \textrm{ is everywhere defined and } D _ \xi f \in X} \} $$

and acting according to the rule $ Af = D _ \xi f $ if $ f \in D(A) $, is called a differentiation operator; $ A $ is a linear operator.

7) Let $ X $ be a vector space of functions on a set $ M $; an everywhere-defined operator assigning to a function $ \phi \in X $ the value of that function at a point $ a \in M $, is a linear functional on $ X $; it is called the $ \delta $- function at the point $ a $ and is written as $ \delta _{a} $.

8) Let $ G $ be a commutative locally compact group, let $ \widehat{G} $ be the group of characters of the group $ G $, let $ dg $, $ \widehat{dg} $ be the Haar measures on $ G $ and $ \widehat{G} $, respectively, and let

$$ X = L _{2} ( G ,\ dg ), Y = L _{2} ( \widehat{G} ,\ \widehat{dg} ). $$

The linear operator $ A $ from $ X $ into $ Y $ assigning to a function $ f \in X $ the function $ \widehat{f} \in Y $ defined by the formula

$$ \widehat{f} ( \widehat{g} ) = \int\limits f(g) \widehat{g} (g) \ dg $$

is everywhere defined if the convergence of the integral is taken to be mean-square convergence.

If $ X $ and $ Y $ are topological vector spaces, then the operators in examples 1) and 2) are continuous; if in example 3) the space $ X $ is $ L _{2} (M,\ \Sigma _{M} ,\ \mu ) $, where $ \mu $ is a measure on $ X $, then the operator of multiplication by a bounded measurable function is closed and has a dense domain of definition; if in example 5) the space $ X=Y $ is a Hilbert space $ L _{2} (M,\ \Sigma _{M} ,\ \mu ) $ and $ K(x,\ y,\ z) = K(x,\ y)z $, where $ K(x,\ y) $ belongs to $ L _{2} (M \times M,\ \Sigma _{M} \times \Sigma _{M} ,\ \mu \times \mu ) $, then $ A $ is compact; if in example 8) the spaces $ X $ and $ Y $ are regarded as Hilbert spaces, then $ A $ is continuous.

If $ A $ is an operator from $ X $ into $ Y $ such that $ Ax \neq Ay $ when $ x \neq y $, $ x,\ y \in D(A) $, then the inverse operator $ A ^{-1} $ to $ A $ can be defined; the question of the existence of an inverse operator and its properties is related to the theorem of the existence and uniqueness of a solution of the equation $ Ax = f $; if $ A ^{-1} $ exists, then $ x = A ^{-1} f $ when $ f \in R(A) $.

For operators on a vector space it is possible to define a sum, multiplication by a number and an operator product. If $ A $, $ B $ are operators from $ X $ into $ Y $ with domains of definition $ D(A) $ and $ D(B) $, respectively, then the operator, written as $ A+B $, with domain of definition

$$ D(A+B) = D(A) \cap D(B) $$

and acting according to the rule

$$ (A+B)x = Ax + Bx $$

if $ x \in D(A+B) $, is called the sum of the operators $ A $ and $ B $.

The operator, written as $ \lambda A $, with domain of definition

$$ D( \lambda A) = D(A) $$

and acting according to the rule

$$ ( \lambda A)x = \lambda (Ax) $$

if $ x \in D( \lambda A) $, is called the product of the operator $ A $ by the number $ \lambda $. The operator product is defined as composition of mappings: If $ A $ is an operator from $ X $ into $ Y $ and $ B $ is an operator from $ Y $ into $ Z $, then the operator $ BA $, with domain of definition

$$ D(BA) = \{ {x \in X} : { x \in D(A) \textrm{ and } Ax \in D(B)} \} $$

and acting according to the rule

$$ (BA)x = B(Ax) $$

if $ x \in D(BA) $, is called the product of $ B $ and $ A $.

If $ P $ is an everywhere-defined operator on $ X $ such that $ PP = P $, then $ P $ is called a projection operator or projector in $ X $; if $ I $ is an everywhere-defined operator on $ X $ such that $ I \circ I = \mathop{\rm id}\nolimits _{X} $, then $ I $ is called an involution in $ X $.

The theory of operators constitutes the most important part of linear and non-linear functional analysis, being in particular a basic instrument in the theory of dynamical systems, representations of groups and algebras and a most important mathematical instrument in mathematical physics and quantum mechanics.

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