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

• 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

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

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.

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.

References

[1]	L.A. [L.A. Lyusternik] Liusternik, "Elements of functional analysis" , F. Ungar (1961) (Translated from Russian)
[2]	A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian)
[3]	L.V. Kantorovich, G.P. Akilov, "Functional analysis in normed spaces" , Pergamon (1964) (Translated from Russian)
[4]	N. Dunford, J.T. Schwartz, "Linear operators" , 1–3 , Interscience (1958)
[5]	R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)
[6]	K. Yosida, "Functional analysis" , Springer (1980)

References

[a1]	T. Kato, "Perturbation theory for linear operators" , Springer (1976)
[a2]	A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980) pp. Chapt. 5
[a3]	F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)
[a4]	W. Rudin, "Functional analysis" , McGraw-Hill (1973)
[a5]	I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)