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A closed [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n0675701.png" /> defined on a linear subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n0675702.png" /> that is dense in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n0675703.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n0675704.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n0675705.png" /> is the operator adjoint to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n0675706.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n0675707.png" /> is normal, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n0675708.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n0675709.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757010.png" />. Conversely, these conditions guarantee that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757011.png" /> is normal. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757012.png" /> is normal, then so are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757013.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757014.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757015.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757016.png" /> when it exists; and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757018.png" /> is a bounded linear operator, then also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757019.png" />.
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A closed [[Linear operator|linear operator]]  $  A $
 +
defined on a linear subspace $  D _ {A} $
 +
that is dense in a Hilbert space $  H $
 +
such that $  A  ^ {*} A = AA  ^ {*} $,  
 +
where $  A  ^ {*} $
 +
is the operator adjoint to $  A $.  
 +
If $  A $
 +
is normal, then $  D _ {A  ^ {*}  } = D _ {A} $
 +
and $  \| A  ^ {*} x \| = \| A x \| $
 +
for every $  x $.  
 +
Conversely, these conditions guarantee that $  A $
 +
is normal. If $  A $
 +
is normal, then so are $  A  ^ {*} $;  
 +
$  \alpha A + \beta I $
 +
for any $  \alpha , \beta \in \mathbf C $;  
 +
$  A  ^ {-} 1 $
 +
when it exists; and if $  AB = BA $,  
 +
where $  B $
 +
is a bounded linear operator, then also $  A  ^ {*} B = BA  ^ {*} $.
  
 
A normal operator has:
 
A normal operator has:
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1) the multiplicative decomposition
 
1) the multiplicative decomposition
  
<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/n/n067/n067570/n06757020.png" /></td> </tr></table>
+
$$
 +
A = U \sqrt {A  ^ {*} A }  = \sqrt {A  ^ {*} A } U ,
 +
$$
  
<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/n/n067/n067570/n06757021.png" /></td> </tr></table>
+
$$
 +
A  ^ {*}  = U  ^ {-} 1 \sqrt {A  ^ {*} A }  = \sqrt {A  ^ {*} A } U  ^ {-} 1 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757022.png" /> is a unitary operator which is uniquely determined on the orthogonal complement of the null space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757024.png" />;
+
where $  U $
 +
is a unitary operator which is uniquely determined on the orthogonal complement of the null space of $  A $
 +
and $  A  ^ {*} $;
  
 
2) the additive decomposition
 
2) the additive decomposition
  
<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/n/n067/n067570/n06757025.png" /></td> </tr></table>
+
$$
 +
= A _ {1} + iA _ {2} ,\ \
 +
A  ^ {*}  = A _ {1} - iA _ {2} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757027.png" /> are uniquely determined self-adjoint commuting operators.
+
where $  A _ {1} $
 +
and $  A _ {2} $
 +
are uniquely determined self-adjoint commuting operators.
  
The additive decomposition implies that for an ordered pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757028.png" /> there exists a unique two-dimensional [[Spectral function|spectral function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757030.png" /> is a two-dimensional interval, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757032.png" />, such that
+
The additive decomposition implies that for an ordered pair $  ( A, A  ^ {*} ) $
 +
there exists a unique two-dimensional [[Spectral function|spectral function]] $  E ( \Delta _  \zeta  ) $,  
 +
where $  \Delta _  \zeta  $
 +
is a two-dimensional interval, $  \Delta _  \zeta  = \Delta _  \xi  \times \Delta _  \eta  $,  
 +
$  \zeta = \xi + i \eta $,  
 +
such that
  
<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/n/n067/n067570/n06757033.png" /></td> </tr></table>
+
$$
 +
= \int\limits _ {\Delta _  \infty  } \zeta  dE ( \Delta _  \zeta  ),\ \
 +
A  ^ {*}  = \int\limits _ {\Delta _  \infty  } \overline \zeta \; dE (
 +
\Delta _  \zeta  ).
 +
$$
  
The same decomposition also implies that a normal operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757034.png" /> is a function of a certain self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757036.png" />. Conversely, every function of some self-adjoint operator is normal.
+
The same decomposition also implies that a normal operator $  A $
 +
is a function of a certain self-adjoint operator $  C $,  
 +
$  A = F ( C) $.  
 +
Conversely, every function of some self-adjoint operator is normal.
  
An important property of a normal operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757037.png" /> is the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757038.png" />, which implies that the [[Spectral radius|spectral radius]] of a normal operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757039.png" /> is its norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067570/n06757040.png" />. Eigen elements of a normal operator corresponding to distinct eigen values are orthogonal.
+
An important property of a normal operator $  A $
 +
is the fact that $  \| A  ^ {n} \| = \| A \|  ^ {n} $,  
 +
which implies that the [[Spectral radius|spectral radius]] of a normal operator $  A $
 +
is its norm $  \| A \| $.  
 +
Eigen elements of a normal operator corresponding to distinct eigen values are orthogonal.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Plesner,  "Spectral theory of linear operators" , F. Ungar  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Rudin,  "Functional analysis" , McGraw-Hill  (1973)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Plesner,  "Spectral theory of linear operators" , F. Ungar  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Rudin,  "Functional analysis" , McGraw-Hill  (1973)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.B. Conway,  "Subnormal operators" , Pitman  (1981)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.B. Conway,  "Subnormal operators" , Pitman  (1981)</TD></TR></table>

Latest revision as of 08:03, 6 June 2020


A closed linear operator $ A $ defined on a linear subspace $ D _ {A} $ that is dense in a Hilbert space $ H $ such that $ A ^ {*} A = AA ^ {*} $, where $ A ^ {*} $ is the operator adjoint to $ A $. If $ A $ is normal, then $ D _ {A ^ {*} } = D _ {A} $ and $ \| A ^ {*} x \| = \| A x \| $ for every $ x $. Conversely, these conditions guarantee that $ A $ is normal. If $ A $ is normal, then so are $ A ^ {*} $; $ \alpha A + \beta I $ for any $ \alpha , \beta \in \mathbf C $; $ A ^ {-} 1 $ when it exists; and if $ AB = BA $, where $ B $ is a bounded linear operator, then also $ A ^ {*} B = BA ^ {*} $.

A normal operator has:

1) the multiplicative decomposition

$$ A = U \sqrt {A ^ {*} A } = \sqrt {A ^ {*} A } U , $$

$$ A ^ {*} = U ^ {-} 1 \sqrt {A ^ {*} A } = \sqrt {A ^ {*} A } U ^ {-} 1 , $$

where $ U $ is a unitary operator which is uniquely determined on the orthogonal complement of the null space of $ A $ and $ A ^ {*} $;

2) the additive decomposition

$$ A = A _ {1} + iA _ {2} ,\ \ A ^ {*} = A _ {1} - iA _ {2} , $$

where $ A _ {1} $ and $ A _ {2} $ are uniquely determined self-adjoint commuting operators.

The additive decomposition implies that for an ordered pair $ ( A, A ^ {*} ) $ there exists a unique two-dimensional spectral function $ E ( \Delta _ \zeta ) $, where $ \Delta _ \zeta $ is a two-dimensional interval, $ \Delta _ \zeta = \Delta _ \xi \times \Delta _ \eta $, $ \zeta = \xi + i \eta $, such that

$$ A = \int\limits _ {\Delta _ \infty } \zeta dE ( \Delta _ \zeta ),\ \ A ^ {*} = \int\limits _ {\Delta _ \infty } \overline \zeta \; dE ( \Delta _ \zeta ). $$

The same decomposition also implies that a normal operator $ A $ is a function of a certain self-adjoint operator $ C $, $ A = F ( C) $. Conversely, every function of some self-adjoint operator is normal.

An important property of a normal operator $ A $ is the fact that $ \| A ^ {n} \| = \| A \| ^ {n} $, which implies that the spectral radius of a normal operator $ A $ is its norm $ \| A \| $. Eigen elements of a normal operator corresponding to distinct eigen values are orthogonal.

References

[1] A.I. Plesner, "Spectral theory of linear operators" , F. Ungar (1965) (Translated from Russian)
[2] W. Rudin, "Functional analysis" , McGraw-Hill (1973)

Comments

References

[a1] J.B. Conway, "Subnormal operators" , Pitman (1981)
How to Cite This Entry:
Normal operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_operator&oldid=48015
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article