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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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A linear operator $A^* : Y^* \rightarrow X^*$ (where $X^*$ and $Y^*$ are the strong duals of [[locally convex space]]s $X$ and $Y$, respectively), constructed from a linear operator $A : X \rightarrow Y$ in the following way. Let the domain of definition $D_A$ of $A$ be everywhere dense in $X$. If for all $x \in D_A$,
 
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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/a/a010/a010870/a01087011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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(Ax, g) = (x, g^*)
 
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\label{1}
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.
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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 (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.
  
 
====References====
 
====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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<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>
  
  
  
 
====Comments====
 
====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
+
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.
  
<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>
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====References====
 +
<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>
  
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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{{TEX|done}}
 
 
====References====
 
<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>
 

Revision as of 20:46, 18 June 2016

A linear operator $A^* : Y^* \rightarrow X^*$ (where $X^*$ and $Y^*$ are the strong duals of locally convex spaces $X$ and $Y$, respectively), constructed from a linear operator $A : X \rightarrow Y$ in the following way. Let the domain of definition $D_A$ of $A$ be everywhere dense in $X$. If for all $x \in D_A$, $$ (Ax, g) = (x, g^*) \label{1} $$ 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 (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.

References

[1] K. Yosida, "Functional analysis" , Springer (1980)
[2] F. Riesz, B. Szökefalvi-Nagy, "Functional analysis" , F. Ungar (1955) (Translated from French)


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

[a1] 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=38989
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article