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A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a0108701.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a0108702.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a0108703.png" /> are the strong duals of locally convex spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a0108704.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a0108705.png" />, respectively), constructed from a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a0108706.png" /> in the following way. Let the domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a0108707.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a0108708.png" /> be everywhere dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a0108709.png" />. If for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087010.png" />,
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{{TEX|done}}
  
<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/a/a010/a010870/a01087011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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A [[linear operator]] $A^*\colon Y^* \rightarrow X^*$ (where $X^*$ and $Y^*$ are the strong duals of [[locally convex space]]s $X$ and $Y$, respectively), constructed from a linear operator $A\colon X \rightarrow Y$ in the following way. Let the domain of definition $D_A$ of $A$ be everywhere dense in $X$. If for all $x \in D_A$,
 +
$$
 +
(Ax, g) = (x, g^*)
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\label{eq:1}
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$$
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where $Ax \in Y$, $g \in Y^*$ and $g^* \in X^*$, then $A^*g = g^*$ is a uniquely defined operator from the set $D_{A^*}$ of elements $g$ satisfying \eqref{eq:1} into $X^*$. If $D_A = X$ and $A$ is [[Continuous operator|continuous]], then $A^*$ is also continuous. If, in addition, $X$ and $Y$ are normed [[linear space]]s, then $\Vert A^* \Vert = \Vert A \Vert$. If $A$ is completely continuous, then so is $A^*$. Adjoint operators are of particular interest in the case when $X$ and $Y$ are Hilbert spaces.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087015.png" /> is a uniquely defined operator from the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087016.png" /> of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087017.png" /> satisfying (*) into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087018.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087020.png" /> is continuous, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087021.png" /> is also continuous. If, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087023.png" /> are normed linear spaces, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087024.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087025.png" /> is completely continuous, then so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087026.png" />. Adjoint operators are of particular interest in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087028.png" /> are Hilbert spaces.
 
  
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  F. Riesz,  B. Szökefalvi-Nagy,  "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR></table>
 
  
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====Comments====
  
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In Western literature the adjoint operator as defined above is usually called the dual or conjugate operator. The term adjoint operator is reserved for [[Hilbert space]]s, in which case it is defined by
 +
$$
 +
(Ax,g) = (x,A^*g)
 +
$$
 +
where $({\cdot},{\cdot})$ denotes the Hilbert space inner product.
  
====Comments====
 
In Western literature the adjoint operator as defined above is usually called the dual or conjugate operator. The term adjoint operator is reserved for Hilbert spaces, in which case it is defined by
 
  
<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/a/a010/a010870/a01087029.png" /></td> </tr></table>
 
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010870/a01087030.png" /> denotes the Hilbert space inner product.
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====References====
  
====References====
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<table>
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.E. Taylor,  D.C. Lay,  "Introduction to functional analysis" , Wiley  (1980)</TD></TR></table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1980)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  F. Riesz,  B. Szökefalvi-Nagy,  "Functional analysis" , F. Ungar  (1955)  (Translated from French)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  A.E. Taylor,  D.C. Lay,  "Introduction to functional analysis" , Wiley  (1980)</TD></TR>
 +
</table>

Latest revision as of 06:42, 21 June 2016


A linear operator $A^*\colon Y^* \rightarrow X^*$ (where $X^*$ and $Y^*$ are the strong duals of locally convex spaces $X$ and $Y$, respectively), constructed from a linear operator $A\colon X \rightarrow Y$ in the following way. Let the domain of definition $D_A$ of $A$ be everywhere dense in $X$. If for all $x \in D_A$, $$ (Ax, g) = (x, g^*) \label{eq:1} $$ where $Ax \in Y$, $g \in Y^*$ and $g^* \in X^*$, then $A^*g = g^*$ is a uniquely defined operator from the set $D_{A^*}$ of elements $g$ satisfying \eqref{eq:1} into $X^*$. If $D_A = X$ and $A$ is continuous, then $A^*$ is also continuous. If, in addition, $X$ and $Y$ are normed linear spaces, then $\Vert A^* \Vert = \Vert A \Vert$. If $A$ is completely continuous, then so is $A^*$. Adjoint operators are of particular interest in the case when $X$ and $Y$ are Hilbert spaces.


Comments

In Western literature the adjoint operator as defined above is usually called the dual or conjugate operator. The term adjoint operator is reserved for Hilbert spaces, in which case it is defined by $$ (Ax,g) = (x,A^*g) $$ where $({\cdot},{\cdot})$ denotes the Hilbert space inner product.


References

[1] K. Yosida, "Functional analysis" , Springer (1980)
[2] F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)
[3] A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980)
How to Cite This Entry:
Adjoint operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_operator&oldid=16759
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article