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− | 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|mapping]] or [[Function|function]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o0683501.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o0683502.png" /> be two sets. A rule or correspondence which assigns a uniquely defined element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o0683503.png" /> to every element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o0683504.png" /> of a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o0683505.png" /> is called an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o0683506.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o0683507.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o0683508.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o0683509.png" /> is called the domain of definition of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835010.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835011.png" />; the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835012.png" /> is called the domain of values of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835013.png" /> (or its range) and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835014.png" />. The expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835015.png" /> is often written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835016.png" />. The term operator is mostly used in the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835018.png" /> are vector spaces. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835019.png" /> is an operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835020.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835021.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835022.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835023.png" /> is called an operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835024.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835025.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835026.png" /> is called an everywhere-defined operator. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835028.png" /> are operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835029.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835030.png" /> and from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835031.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835032.png" /> with domains of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835034.png" />, respectively, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835036.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835037.png" />, then if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835039.png" />, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835040.png" /> is called a compression or restriction of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835041.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835042.png" /> is called an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835043.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835045.png" /> is called an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835046.png" /> exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835047.png" />.
| + | {{TEX|done}} |
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− | Many equations in function spaces or abstract spaces can be expressed in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835050.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835051.png" /> is given, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835052.png" /> is unknown and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835053.png" /> is an operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835054.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835055.png" />. The assertion of the existence of a solution to this equation for any right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835056.png" /> is equivalent to the assertion that the range of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835057.png" /> is the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835058.png" />; the assertion that the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835059.png" /> has a unique solution for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835060.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835061.png" /> is a one-to-one mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835062.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835063.png" />.
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− | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835065.png" /> are vector spaces, then in the set of all operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835066.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835067.png" /> it is possible to single out the class of linear operators (cf. [[Linear operator|Linear operator]]); the remaining operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835068.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835069.png" /> are called non-linear operators. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835071.png" /> are topological vector spaces, then in the set of operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835072.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835073.png" /> the class of continuous operators (cf. [[Continuous operator|Continuous operator]]) can be naturally singled out, so are the class of bounded linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835074.png" /> (operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835075.png" /> such that the image of any bounded set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835076.png" /> is bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835077.png" />) and the class of compact linear operators (i.e. operators such that the image of any bounded set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835078.png" /> is pre-compact in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835079.png" />, cf. [[Compact operator|Compact operator]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835081.png" /> are locally convex spaces, then it is natural to examine different topologies on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835083.png" />; an operator is said to be semi-continuous if it defines a continuous mapping from the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835084.png" /> (with the initial topology) into the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835085.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835086.png" /> with the boundedly weak topology into the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835087.png" />; an operator is called weakly continuous if it defines a continuous mapping from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835088.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835089.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835091.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835093.png" /> 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.
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− | The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835094.png" /> defined by the relation
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835095.png" /></td> </tr></table>
| + | 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|mapping]] or [[Function|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|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|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|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. |
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− | is called the graph of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835096.png" />. | + | ===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 |
| + | $\operatorname{R}(A)$. |
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− | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835097.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835098.png" /> be topological vector spaces; an operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o06835099.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350100.png" /> is called a [[Closed operator|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.
| + | ===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|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. |
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− | The concept of a graph allows one to generalize the concept of an operator: Any subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350101.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350102.png" /> is called a multi-valued operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350103.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350104.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350106.png" /> are vector spaces, then a linear subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350107.png" /> is called a multi-valued linear operator; the set | + | 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|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|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.=== |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350108.png" /></td> </tr></table>
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350109.png" /></td> </tr></table>
| + | 1) The operator assigning the element $ 0 \in Y $ |
| + | to any element $ x \in X $( |
| + | the zero operator). |
| | | |
− | is called the domain of definition of the multi-valued 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 |
| | | |
− | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350110.png" /> is a vector space over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350111.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350112.png" />, then an everywhere-defined operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350113.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350114.png" /> is called a [[Functional|functional]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350115.png" />.
| + | $$ |
| + | D(A) = \{ {\phi \in X} : {f \phi \in X} \} |
| + | $$ |
| | | |
− | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350117.png" /> are locally convex spaces, then an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350118.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350119.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350120.png" /> with a dense domain of definition in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350121.png" /> has an [[Adjoint operator|adjoint operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350122.png" /> with a dense domain of definition in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350123.png" /> (with the weak topology) if, and only if, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350124.png" /> is a closed operator.
| |
| | | |
− | ===Examples of operators.=== | + | and acting according to the rule |
| + | |
| + | $$ |
| + | A \phi = f \phi |
| + | $$ |
| | | |
| | | |
− | 1) The operator assigning the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350125.png" /> to any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350126.png" /> (the zero operator).
| + | if $ \phi \in D(A) $, |
| + | is called the operator of multiplication by a function; $ A $ |
| + | is a linear operator. |
| | | |
− | 2) The operator mapping each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350127.png" /> to the same element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350128.png" /> (the identity operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350129.png" />, written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350130.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350131.png" />).
| + | 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 |
| | | |
− | 3) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350132.png" /> be a vector space of functions on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350133.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350134.png" /> be a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350135.png" />; the operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350136.png" /> with domain of definition
| + | $$ |
| + | D(A) = \{ {\phi \in X} : {\phi \circ F \in X} \} |
| + | $$ |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350137.png" /></td> </tr></table>
| |
| | | |
| and acting according to the rule | | and acting according to the rule |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350138.png" /></td> </tr></table>
| + | $$ |
| + | A \phi = \phi \circ F |
| + | $$ |
| | | |
− | if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350139.png" />, is called the operator of multiplication by a function; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350140.png" /> is a linear operator.
| |
| | | |
− | 4) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350141.png" /> be a vector space of functions on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350142.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350143.png" /> be a mapping from the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350144.png" /> into itself; the operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350145.png" /> with domain of definition
| + | if $ \phi \in D(A) $, |
| + | is a linear operator. |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350146.png" /></td> </tr></table>
| + | 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 |
| | | |
− | and acting according to the rule
| + | $$ |
| + | K(x,\ y,\ z) = K(x,\ y)z, x \in M, y \in N, z \in \mathbf R , |
| + | $$ |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350147.png" /></td> </tr></table>
| |
| | | |
− | if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350148.png" />, is a linear operator.
| + | then $ A $ |
| + | is a linear operator. |
| | | |
− | 5) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350149.png" /> be vector spaces of real measurable functions on two measure spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350150.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350151.png" />, respectively, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350152.png" /> be a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350153.png" />, measurable with respect to the product measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350154.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350155.png" /> is Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350156.png" />, and continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350157.png" /> for any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350158.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350159.png" />. The operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350160.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350161.png" /> with domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350162.png" />, which exists for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350163.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350164.png" />, and acting according to the rule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350165.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350166.png" />, is called an integral operator; if
| + | 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 |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350167.png" /></td> </tr></table>
| + | $$ |
| + | D(A) = |
| + | \{ {f \in X} : {\textrm{ the derivative } D _ \xi f \textrm{ of the function } f |
| + | \textrm{ along the field } \xi \textrm{ is everywhere defined and } D _ \xi f \in X} \} |
| + | $$ |
| | | |
− | then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350168.png" /> is a linear operator.
| |
| | | |
− | 6) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350169.png" /> be a vector space of functions on a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350170.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350171.png" /> be a vector field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350172.png" />; the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350173.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350174.png" /> with domain of definition
| + | 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. |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350175.png" /></td> </tr></table>
| + | 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} $. |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350176.png" /></td> </tr></table>
| |
| | | |
− | and acting according to the rule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350177.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350178.png" />, is called a differentiation operator; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350179.png" /> is a linear operator. | + | 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 |
| | | |
− | 7) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350180.png" /> be a vector space of functions on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350181.png" />; an everywhere-defined operator assigning to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350182.png" /> the value of that function at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350183.png" />, is a linear functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350184.png" />; it is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350186.png" />-function at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350187.png" /> and is written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350188.png" />.
| + | $$ |
| + | X = L _{2} ( G ,\ dg ), |
| + | Y = L _{2} ( \widehat{G} ,\ \widehat{dg} ). |
| + | $$ |
| | | |
− | 8) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350189.png" /> be a commutative locally compact group, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350190.png" /> be the group of characters of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350191.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350192.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350193.png" /> be the Haar measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350194.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350195.png" />, respectively, and let
| |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350196.png" /></td> </tr></table>
| + | 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 |
| | | |
− | The linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350197.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350198.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350199.png" /> assigning to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350200.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350201.png" /> defined by the formula
| + | $$ |
| + | \widehat{f} ( \widehat{g} ) = \int\limits f(g) \widehat{g} (g) \ dg |
| + | $$ |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350202.png" /></td> </tr></table>
| |
| | | |
| is everywhere defined if the convergence of the integral is taken to be mean-square convergence. | | is everywhere defined if the convergence of the integral is taken to be mean-square convergence. |
| | | |
− | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350203.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350204.png" /> are topological vector spaces, then the operators in examples 1) and 2) are continuous; if in example 3) the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350205.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350206.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350207.png" /> is a measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350208.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350209.png" /> is a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350210.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350211.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350212.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350213.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350214.png" /> is compact; if in example 8) the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350215.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350216.png" /> are regarded as Hilbert spaces, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350217.png" /> is continuous. | + | 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) $. |
| + | |
| | | |
− | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350218.png" /> is an operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350219.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350220.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350221.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350222.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350223.png" />, then the inverse operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350224.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350225.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350226.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350227.png" /> exists, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350228.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350229.png" />.
| + | 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 |
| | | |
− | For operators on a vector space it is possible to define a sum, multiplication by a number and an operator product. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350230.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350231.png" /> are operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350232.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350233.png" /> with domains of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350234.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350235.png" />, respectively, then the operator, written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350236.png" />, with domain of definition
| + | $$ |
| + | D(A+B) = D(A) \cap D(B) |
| + | $$ |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350237.png" /></td> </tr></table>
| |
| | | |
| and acting according to the rule | | and acting according to the rule |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350238.png" /></td> </tr></table>
| + | $$ |
| + | (A+B)x = Ax + Bx |
| + | $$ |
| + | |
| | | |
− | if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350239.png" />, is called the sum of the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350240.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350241.png" />. | + | if $ x \in D(A+B) $, |
| + | is called the sum of the operators $ A $ |
| + | and $ B $. |
| | | |
− | The operator, written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350242.png" />, with domain of definition
| |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350243.png" /></td> </tr></table>
| + | The operator, written as $ \lambda A $, |
| + | with domain of definition |
| + | |
| + | $$ |
| + | D( \lambda A) = D(A) |
| + | $$ |
| + | |
| | | |
| and acting according to the rule | | and acting according to the rule |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350244.png" /></td> </tr></table>
| + | $$ |
| + | ( \lambda A)x = \lambda (Ax) |
| + | $$ |
| | | |
− | if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350245.png" />, is called the product of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350246.png" /> by the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350247.png" />. The operator product is defined as composition of mappings: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350248.png" /> is an operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350249.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350250.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350251.png" /> is an operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350252.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350253.png" />, then the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350254.png" />, with domain of definition
| |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350255.png" /></td> </tr></table>
| + | 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 | | and acting according to the rule |
| | | |
− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350256.png" /></td> </tr></table>
| + | $$ |
| + | (BA)x = B(Ax) |
| + | $$ |
| + | |
| + | |
| + | if $ x \in D(BA) $, |
| + | is called the product of $ B $ |
| + | and $ A $. |
| + | |
| | | |
− | if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350257.png" />, is called the product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350258.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350259.png" />. | + | 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 $. |
| | | |
− | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350260.png" /> is an everywhere-defined operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350261.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350262.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350263.png" /> is called a projection operator or projector in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350264.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350265.png" /> is an everywhere-defined operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350266.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350267.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350268.png" /> is called an involution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o068/o068350/o068350269.png" />.
| |
| | | |
| 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. | | 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. |