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− | A linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s0916901.png" /> of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s0916902.png" /> in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s0916903.png" /> (in general, complex) into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s0916904.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s0916905.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s0916906.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s0916907.png" /> is an everywhere-dense linear manifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s0916908.png" /> (and this is assumed in what follows), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s0916909.png" /> is a linear operator. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169011.png" /> is bounded and hence continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169012.png" />. A symmetric operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169013.png" /> induces a bilinear Hermitian form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169014.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169015.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169016.png" />. The corresponding quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169017.png" /> is real. Conversely, if the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169019.png" /> is real, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169020.png" /> is symmetric. The sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169021.png" /> of two symmetric operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169023.png" /> with a common domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169024.png" /> is again a symmetric operator, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169025.png" /> is a real number, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169026.png" /> is also symmetric. Every symmetric operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169027.png" /> has a uniquely defined closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169028.png" /> and an adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169029.png" />. In general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169030.png" /> is not symmetric and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169031.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169032.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169033.png" /> is called a [[Self-adjoint operator|self-adjoint operator]]. This holds, for example, in the case of symmetric operators defined on the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169034.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169035.png" /> is symmetric and bounded on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169036.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169037.png" /> can be extended as a bounded symmetric operator to the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169038.png" />.
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− | ===Examples.=== | + | A linear [[mapping]] $A$ of a set $D_A$ in a [[Hilbert space]] $H$ (in general, complex) into $H$ such that $\langle Ax,y\rangle =\langle x,yA\rangle$ for all $x,y\in D_A$. If $D_A$ is an everywhere-dense linear [[manifold]] in $H$ (and this is assumed in what follows), then $A$ is a [[linear operator]]. If $D_A=H$, then $A$ is bounded and hence continuous on $H$. A symmetric operator $A$ induces a bilinear [[Hermitian form]] $B(x,y)=\langle Ax,y\rangle$ on $D_A$, that is, $B(x,y)=\overline{B(x,y)}$. The corresponding quadratic form $\langle Ax,x\rangle$ is real. Conversely, if the form $\langle Ax,x\rangle$ on $D_A$ is real, then $A$ is symmetric. The sum $A+B$ of two symmetric operators $A$ and $B$ with a common [[domain]] of definition $D_A=D_B$ is again a symmetric operator, while if $\lambda$ is a real number, then $\lambda A$ is also symmetric. Every symmetric operator $A$ has a uniquely defined closure $\overline{A}$ and an adjoint $A^* \supset\overline{A}$. In general, $A^*$ is not symmetric and $A^*\neq\overline{A}$. If $A^*=A$, then $A$ is called a [[self-adjoint operator]]. This holds, for example, in the case of symmetric operators defined on the whole of $H$. If $A$ is symmetric and bounded on $D_A$, then $A$ can be extended as a bounded symmetric operator to the whole of $H$. |
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− | 1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169040.png" /> be an infinite matrix such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169041.png" />, and
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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/s/s091/s091690/s09169042.png" /></td> </tr></table>
| + | ===Examples=== |
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− | Then the system of equations | + | # Let $\|a_{ij}\|$, $i,j=1,2,\ldots$, be an infinite matrix such that $a_{ij}=\overline{a}_{ji}$, and \begin{equation}\sum_{i,j=1}^{\infty}|a_{ij}|^2<\infty .\end{equation} Then the system of equations \begin{equation}\eta_i=\sum_{j=1}^{\infty}a_{ij}\xi_j,\quad i=1,2,\ldots,\end{equation} defining $y=\{\eta_i\}$ for an $x=\{\xi_i\}\in l_2$, defines a bounded symmetric operator, which turns out to be self-adjoint on the complex space $l_2$. |
| + | # In the complex space $L_2(0,1)$, let $A=id/dt$ be defined on the set $D_A$ of [[absolutely continuous function|absolutely-continuous functions]] $x$ on $[0,1]$ having a square-summable derivative and satisfying the condition $x(0)=x(1)=0$. Then $A$ is symmetric but not self-adjoint. |
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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/s/s091/s091690/s09169043.png" /></td> </tr></table>
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− | defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169044.png" /> for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169045.png" />, defines a bounded symmetric operator, which turns out to be self-adjoint on the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169046.png" />.
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− | 2) In the complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169047.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169048.png" /> be defined on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169049.png" /> of absolutely-continuous functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169050.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169051.png" /> having a square-summable derivative and satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169052.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091690/s09169053.png" /> is symmetric but not self-adjoint.
| + | ===References=== |
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− | ====References====
| + | <table> |
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.A. [L.A. Lyusternik] Liusternik, "Elements of functional analysis" , F. Ungar (1961) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> F. Riesz, B. Szökevalfi-Nagy, "Leçons d'analyse fonctionelle" , Akad. Kiado (1955)</TD></TR></table> | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> L.A. [L.A. Lyusternik] Liusternik, "Elements of functional analysis" , F. Ungar (1961) (Translated from Russian)</TD></TR> |
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> F. Riesz, B. Szökevalfi-Nagy, "Leçons d'analyse fonctionelle" , Akad. Kiado (1955)</TD></TR> |
| + | </table> |
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− | ====Comments====
| + | ===Comments=== |
| + | |
| An important problem is to find a self-adjoint extension of a symmetric operator. This problem has different versions, depending on whether one looks for an extension in the original or in a larger space. A complete theory of this topic exists. | | An important problem is to find a self-adjoint extension of a symmetric operator. This problem has different versions, depending on whether one looks for an extension in the original or in a larger space. A complete theory of this topic exists. |
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− | ====References====
| + | |
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman (1980) (Translated from Russian)</TD></TR></table> | + | |
| + | ===References=== |
| + | |
| + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , '''1–2''' , Pitman (1980) (Translated from Russian)</TD></TR> |
| + | </table> |
A linear mapping $A$ of a set $D_A$ in a Hilbert space $H$ (in general, complex) into $H$ such that $\langle Ax,y\rangle =\langle x,yA\rangle$ for all $x,y\in D_A$. If $D_A$ is an everywhere-dense linear manifold in $H$ (and this is assumed in what follows), then $A$ is a linear operator. If $D_A=H$, then $A$ is bounded and hence continuous on $H$. A symmetric operator $A$ induces a bilinear Hermitian form $B(x,y)=\langle Ax,y\rangle$ on $D_A$, that is, $B(x,y)=\overline{B(x,y)}$. The corresponding quadratic form $\langle Ax,x\rangle$ is real. Conversely, if the form $\langle Ax,x\rangle$ on $D_A$ is real, then $A$ is symmetric. The sum $A+B$ of two symmetric operators $A$ and $B$ with a common domain of definition $D_A=D_B$ is again a symmetric operator, while if $\lambda$ is a real number, then $\lambda A$ is also symmetric. Every symmetric operator $A$ has a uniquely defined closure $\overline{A}$ and an adjoint $A^* \supset\overline{A}$. In general, $A^*$ is not symmetric and $A^*\neq\overline{A}$. If $A^*=A$, then $A$ is called a self-adjoint operator. This holds, for example, in the case of symmetric operators defined on the whole of $H$. If $A$ is symmetric and bounded on $D_A$, then $A$ can be extended as a bounded symmetric operator to the whole of $H$.
Examples
- Let $\|a_{ij}\|$, $i,j=1,2,\ldots$, be an infinite matrix such that $a_{ij}=\overline{a}_{ji}$, and \begin{equation}\sum_{i,j=1}^{\infty}|a_{ij}|^2<\infty .\end{equation} Then the system of equations \begin{equation}\eta_i=\sum_{j=1}^{\infty}a_{ij}\xi_j,\quad i=1,2,\ldots,\end{equation} defining $y=\{\eta_i\}$ for an $x=\{\xi_i\}\in l_2$, defines a bounded symmetric operator, which turns out to be self-adjoint on the complex space $l_2$.
- In the complex space $L_2(0,1)$, let $A=id/dt$ be defined on the set $D_A$ of absolutely-continuous functions $x$ on $[0,1]$ having a square-summable derivative and satisfying the condition $x(0)=x(1)=0$. Then $A$ is symmetric but not self-adjoint.
References
[1] | L.A. [L.A. Lyusternik] Liusternik, "Elements of functional analysis" , F. Ungar (1961) (Translated from Russian) |
[2] | F. Riesz, B. Szökevalfi-Nagy, "Leçons d'analyse fonctionelle" , Akad. Kiado (1955) |
An important problem is to find a self-adjoint extension of a symmetric operator. This problem has different versions, depending on whether one looks for an extension in the original or in a larger space. A complete theory of this topic exists.
References
[a1] | N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , 1–2 , Pitman (1980) (Translated from Russian) |