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A polar decomposition of a linear transformation on a finite-dimensional Euclidean (or unitary) space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p0734301.png" /> is a decomposition of the [[Linear transformation|linear transformation]] into a product of a self-adjoint and an orthogonal (respectively, unitary) transformation (cf. [[Orthogonal transformation|Orthogonal transformation]]; [[Self-adjoint linear transformation|Self-adjoint linear transformation]]; [[Unitary transformation|Unitary transformation]]). Any linear transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p0734302.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p0734303.png" /> has a polar decomposition
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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/p/p073/p073430/p0734304.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p0734305.png" /> is a positive semi-definite self-adjoint linear transformation and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p0734306.png" /> is an orthogonal (or unitary) linear transformation; moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p0734307.png" /> is uniquely defined. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p0734308.png" /> is non-degenerate, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p0734309.png" /> is even positive definite and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343010.png" /> is also uniquely defined. A polar decomposition on a one-dimensional unitary space coincides with the trigonometric representation of a complex number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343011.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343012.png" />.
+
A polar decomposition of a linear transformation on a finite-dimensional Euclidean (or unitary) space  $  L $
 +
is a decomposition of the [[Linear transformation|linear transformation]] into a product of a self-adjoint and an orthogonal (respectively, unitary) transformation (cf. [[Orthogonal transformation|Orthogonal transformation]]; [[Self-adjoint linear transformation|Self-adjoint linear transformation]]; [[Unitary transformation|Unitary transformation]]). Any linear transformation  $  A $
 +
on  $  L $
 +
has a polar decomposition
 +
 
 +
$$
 +
A  =  S \cdot U ,
 +
$$
 +
 
 +
where  $  S $
 +
is a positive semi-definite self-adjoint linear transformation and $  U $
 +
is an orthogonal (or unitary) linear transformation; moreover, $  S $
 +
is uniquely defined. If $  A $
 +
is non-degenerate, then $  S $
 +
is even positive definite and $  U $
 +
is also uniquely defined. A polar decomposition on a one-dimensional unitary space coincides with the trigonometric representation of a complex number $  z $
 +
as $  z = re ^ {i \phi } $.
  
 
''A.L. Onishchik''
 
''A.L. Onishchik''
  
A polar decomposition of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343013.png" /> acting on a [[Hilbert space|Hilbert space]] is a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343014.png" /> in the form
+
A polar decomposition of an operator $  A $
 +
acting on a [[Hilbert space|Hilbert space]] is a representation of $  A $
 +
in the form
  
<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/p/p073/p073430/p07343015.png" /></td> </tr></table>
+
$$
 +
= U T,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343016.png" /> is a partial [[Isometric operator|isometric operator]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343017.png" /> is a [[Positive operator|positive operator]]. Any closed operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343018.png" /> has a polar decomposition, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343019.png" /> (which is often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343020.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343021.png" /> maps the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343022.png" /> of the domain of the self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343023.png" /> into the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343024.png" /> of the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343025.png" /> (the von Neumann theorem, see ). A polar decomposition becomes unique if the source and target subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343026.png" /> are required to coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343028.png" />, respectively. On the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343029.png" /> can be always chosen unitary, isometric or co-isometric, depending on the relation between the codimensions of the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343031.png" />. In particular, if
+
where $  U $
 +
is a partial [[Isometric operator|isometric operator]] and $  T $
 +
is a [[Positive operator|positive operator]]. Any closed operator $  A $
 +
has a polar decomposition, moreover, $  T = ( A  ^ {*} A )  ^ {1/2} $(
 +
which is often denoted by $  T = | A | $),  
 +
and $  U $
 +
maps the closure $  \overline{R}\; _ {A  ^ {*}  } $
 +
of the domain of the self-adjoint operator $  A $
 +
into the closure $  \overline{R}\; _ {A} $
 +
of the range of $  A $(
 +
the von Neumann theorem, see ). A polar decomposition becomes unique if the source and target subspaces of $  U $
 +
are required to coincide with $  \overline{R}\; _ {A  ^ {*}  } $
 +
and $  \overline{R}\; _ {A} $,  
 +
respectively. On the other hand, $  U $
 +
can be always chosen unitary, isometric or co-isometric, depending on the relation between the codimensions of the subspaces $  \overline{R}\; _ {A  ^ {*}  } $
 +
and $  \overline{R}\; _ {A} $.  
 +
In particular, if
  
<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/p/p073/p073430/p07343032.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm dim}  H \ominus \overline{R}\; _ {A  ^ {*}  }  = \
 +
\mathop{\rm dim}  H \ominus \overline{R}\; _ {A} ,
 +
$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343033.png" /> can be chosen unitary and there is a Hermitian operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343034.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343035.png" />. Then the polar decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343036.png" /> takes the form
+
then $  U $
 +
can be chosen unitary and there is a Hermitian operator $  \Phi $
 +
such that $  U = \mathop{\rm exp} ( i \Phi ) $.  
 +
Then the polar decomposition of $  A $
 +
takes the form
  
<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/p/p073/p073430/p07343037.png" /></td> </tr></table>
+
$$
 +
=   \mathop{\rm exp} ( i \Phi )  | A | ,
 +
$$
  
 
entirely analogous to the polar decomposition of a complex number. Commutativity of the terms in a polar decomposition takes place if and only if the operator is normal (cf. [[Normal operator|Normal operator]]).
 
entirely analogous to the polar decomposition of a complex number. Commutativity of the terms in a polar decomposition takes place if and only if the operator is normal (cf. [[Normal operator|Normal operator]]).
Line 23: Line 76:
 
An expression analogous to the polar decomposition has been obtained for operators on a [[Space with an indefinite metric|space with an indefinite metric]] (see , ).
 
An expression analogous to the polar decomposition has been obtained for operators on a [[Space with an indefinite metric|space with an indefinite metric]] (see , ).
  
A polar decomposition of a functional on a von Neumann algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343038.png" /> is a representation of a normal functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343040.png" /> in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343041.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343042.png" /> is a positive normal functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343044.png" /> is a partial isometry (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343046.png" /> are projectors), and multiplication is understood as the action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343047.png" /> of the operator which is adjoint to left multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343048.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343049.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343050.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343051.png" />. A polar decomposition can always be realized so that the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343052.png" /> is fulfilled. Under this condition a polar decomposition is unique.
+
A polar decomposition of a functional on a von Neumann algebra $  A $
 +
is a representation of a normal functional $  f $
 +
on $  A $
 +
in the form $  f = u p $,  
 +
where p $
 +
is a positive normal functional on $  A $,  
 +
$  u \in A $
 +
is a partial isometry (i.e. $  u  ^ {*} u $
 +
and $  u u  ^ {*} $
 +
are projectors), and multiplication is understood as the action on p $
 +
of the operator which is adjoint to left multiplication by $  u $
 +
in $  A $:  
 +
$  f ( x) = p ( u x ) $
 +
for all $  x \in A $.  
 +
A polar decomposition can always be realized so that the condition $  u  ^ {*} f = p $
 +
is fulfilled. Under this condition a polar decomposition is unique.
  
Any bounded linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343053.png" /> on an arbitrary [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343054.png" />-algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343055.png" /> can be considered as a normal functional on the universal enveloping [[Von Neumann algebra|von Neumann algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343056.png" />; the corresponding polar decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343057.png" /> is called the enveloping polar decomposition of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343058.png" />. The restriction of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343059.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343060.png" /> is called the absolute value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343061.png" /> and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343062.png" />; the following properties determine the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343063.png" /> uniquely:
+
Any bounded linear functional $  f $
 +
on an arbitrary [[C*-algebra| $  C  ^ {*} $-
 +
algebra]] $  A $
 +
can be considered as a normal functional on the universal enveloping [[Von Neumann algebra|von Neumann algebra]] $  A  ^ {\prime\prime} $;  
 +
the corresponding polar decomposition $  f = u p $
 +
is called the enveloping polar decomposition of the functional $  f $.  
 +
The restriction of the functional p $
 +
to $  A $
 +
is called the absolute value of $  f $
 +
and is denoted by $  | f | $;  
 +
the following properties determine the functional $  | f | $
 +
uniquely:
  
<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/p/p073/p073430/p07343064.png" /></td> </tr></table>
+
$$
 +
\| | f | \|  = \| f \|
 +
\  \textrm{ and } \  | f ( x) |  ^ {2}  \leq  \| f
 +
\| \cdot | f | ( x  ^ {*} x ) .
 +
$$
  
In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343065.png" /> is the algebra of all continuous functions on a compactum, the absolute value of a functional corresponds to the total variation of the measure determined by it (cf. also [[Total variation of a function|Total variation of a function]]).
+
In the case when $  A = C ( X) $
 +
is the algebra of all continuous functions on a compactum, the absolute value of a functional corresponds to the total variation of the measure determined by it (cf. also [[Total variation of a function|Total variation of a function]]).
  
In many cases a polar decomposition of a functional allows one to reduce studies of functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343066.png" />-algebras to studies of positive functionals. It enables one, for example, to construct for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343067.png" /> a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343068.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343069.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343070.png" /> has a vector realization (i.e. there are vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343071.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343072.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343074.png" />). The representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343075.png" /> constructed from the positive functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343076.png" /> using the GNS-construction (of Gel'fand–Naimark–Segal) has that property.
+
In many cases a polar decomposition of a functional allows one to reduce studies of functionals on $  C  ^ {*} $-
 +
algebras to studies of positive functionals. It enables one, for example, to construct for each $  f \in A  ^  \prime  $
 +
a representation $  \pi $
 +
of the algebra $  A $
 +
on which $  f $
 +
has a vector realization (i.e. there are vectors $  \xi , \eta $
 +
in $  H _  \pi  $
 +
such that $  f ( x) = ( \pi ( x) \xi , \eta ) $,  
 +
$  x \in A $).  
 +
The representation $  \pi _ {| f | }  $
 +
constructed from the positive functional $  | f | $
 +
using the GNS-construction (of Gel'fand–Naimark–Segal) has that property.
  
The polar decomposition of an element of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343078.png" />-algebra is a representation of the element as the product of a positive element and a partial isometric element. Polar decomposition is not valid for all elements: in the usual polar decomposition of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343079.png" /> on a Hilbert space the positive term belongs to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343080.png" />-algebra generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343081.png" />, but for the partial isometric term one can only state that it belongs to the von Neumann algebra generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343082.png" />. That is why one defines and uses the so-called enveloping polar decomposition of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343084.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343085.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343087.png" /> is a partial isometric element in the universal enveloping von Neumann algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343088.png" /> (it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343089.png" /> is canonically imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073430/p07343090.png" />).
+
The polar decomposition of an element of a $  C  ^ {*} $-
 +
algebra is a representation of the element as the product of a positive element and a partial isometric element. Polar decomposition is not valid for all elements: in the usual polar decomposition of an operator $  T $
 +
on a Hilbert space the positive term belongs to the $  C  ^ {*} $-
 +
algebra generated by $  T $,  
 +
but for the partial isometric term one can only state that it belongs to the von Neumann algebra generated by $  T $.  
 +
That is why one defines and uses the so-called enveloping polar decomposition of an element $  a \in A $:  
 +
$  a = u t $,  
 +
where $  t = ( a  ^ {*} a )  ^ {1/2} \in A $
 +
and $  u $
 +
is a partial isometric element in the universal enveloping von Neumann algebra $  A  ^ {\prime\prime} $(
 +
it is assumed that $  A $
 +
is canonically imbedded in $  A  ^ {\prime\prime} $).
  
 
====References====
 
====References====
Line 41: Line 147:
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. [I.Ts. Gokhberg] Gohberg,  M.G. Krein,  "Introduction to the theory of linear nonselfadjoint operators" , ''Transl. Math. Monogr.'' , '''18''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. [I.Ts. Gokhberg] Gohberg,  M.G. Krein,  "Introduction to the theory of linear nonselfadjoint operators" , ''Transl. Math. Monogr.'' , '''18''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>

Latest revision as of 08:06, 6 June 2020


A polar decomposition of a linear transformation on a finite-dimensional Euclidean (or unitary) space $ L $ is a decomposition of the linear transformation into a product of a self-adjoint and an orthogonal (respectively, unitary) transformation (cf. Orthogonal transformation; Self-adjoint linear transformation; Unitary transformation). Any linear transformation $ A $ on $ L $ has a polar decomposition

$$ A = S \cdot U , $$

where $ S $ is a positive semi-definite self-adjoint linear transformation and $ U $ is an orthogonal (or unitary) linear transformation; moreover, $ S $ is uniquely defined. If $ A $ is non-degenerate, then $ S $ is even positive definite and $ U $ is also uniquely defined. A polar decomposition on a one-dimensional unitary space coincides with the trigonometric representation of a complex number $ z $ as $ z = re ^ {i \phi } $.

A.L. Onishchik

A polar decomposition of an operator $ A $ acting on a Hilbert space is a representation of $ A $ in the form

$$ A = U T, $$

where $ U $ is a partial isometric operator and $ T $ is a positive operator. Any closed operator $ A $ has a polar decomposition, moreover, $ T = ( A ^ {*} A ) ^ {1/2} $( which is often denoted by $ T = | A | $), and $ U $ maps the closure $ \overline{R}\; _ {A ^ {*} } $ of the domain of the self-adjoint operator $ A $ into the closure $ \overline{R}\; _ {A} $ of the range of $ A $( the von Neumann theorem, see ). A polar decomposition becomes unique if the source and target subspaces of $ U $ are required to coincide with $ \overline{R}\; _ {A ^ {*} } $ and $ \overline{R}\; _ {A} $, respectively. On the other hand, $ U $ can be always chosen unitary, isometric or co-isometric, depending on the relation between the codimensions of the subspaces $ \overline{R}\; _ {A ^ {*} } $ and $ \overline{R}\; _ {A} $. In particular, if

$$ \mathop{\rm dim} H \ominus \overline{R}\; _ {A ^ {*} } = \ \mathop{\rm dim} H \ominus \overline{R}\; _ {A} , $$

then $ U $ can be chosen unitary and there is a Hermitian operator $ \Phi $ such that $ U = \mathop{\rm exp} ( i \Phi ) $. Then the polar decomposition of $ A $ takes the form

$$ A = \mathop{\rm exp} ( i \Phi ) | A | , $$

entirely analogous to the polar decomposition of a complex number. Commutativity of the terms in a polar decomposition takes place if and only if the operator is normal (cf. Normal operator).

An expression analogous to the polar decomposition has been obtained for operators on a space with an indefinite metric (see , ).

A polar decomposition of a functional on a von Neumann algebra $ A $ is a representation of a normal functional $ f $ on $ A $ in the form $ f = u p $, where $ p $ is a positive normal functional on $ A $, $ u \in A $ is a partial isometry (i.e. $ u ^ {*} u $ and $ u u ^ {*} $ are projectors), and multiplication is understood as the action on $ p $ of the operator which is adjoint to left multiplication by $ u $ in $ A $: $ f ( x) = p ( u x ) $ for all $ x \in A $. A polar decomposition can always be realized so that the condition $ u ^ {*} f = p $ is fulfilled. Under this condition a polar decomposition is unique.

Any bounded linear functional $ f $ on an arbitrary $ C ^ {*} $- algebra $ A $ can be considered as a normal functional on the universal enveloping von Neumann algebra $ A ^ {\prime\prime} $; the corresponding polar decomposition $ f = u p $ is called the enveloping polar decomposition of the functional $ f $. The restriction of the functional $ p $ to $ A $ is called the absolute value of $ f $ and is denoted by $ | f | $; the following properties determine the functional $ | f | $ uniquely:

$$ \| | f | \| = \| f \| \ \textrm{ and } \ | f ( x) | ^ {2} \leq \| f \| \cdot | f | ( x ^ {*} x ) . $$

In the case when $ A = C ( X) $ is the algebra of all continuous functions on a compactum, the absolute value of a functional corresponds to the total variation of the measure determined by it (cf. also Total variation of a function).

In many cases a polar decomposition of a functional allows one to reduce studies of functionals on $ C ^ {*} $- algebras to studies of positive functionals. It enables one, for example, to construct for each $ f \in A ^ \prime $ a representation $ \pi $ of the algebra $ A $ on which $ f $ has a vector realization (i.e. there are vectors $ \xi , \eta $ in $ H _ \pi $ such that $ f ( x) = ( \pi ( x) \xi , \eta ) $, $ x \in A $). The representation $ \pi _ {| f | } $ constructed from the positive functional $ | f | $ using the GNS-construction (of Gel'fand–Naimark–Segal) has that property.

The polar decomposition of an element of a $ C ^ {*} $- algebra is a representation of the element as the product of a positive element and a partial isometric element. Polar decomposition is not valid for all elements: in the usual polar decomposition of an operator $ T $ on a Hilbert space the positive term belongs to the $ C ^ {*} $- algebra generated by $ T $, but for the partial isometric term one can only state that it belongs to the von Neumann algebra generated by $ T $. That is why one defines and uses the so-called enveloping polar decomposition of an element $ a \in A $: $ a = u t $, where $ t = ( a ^ {*} a ) ^ {1/2} \in A $ and $ u $ is a partial isometric element in the universal enveloping von Neumann algebra $ A ^ {\prime\prime} $( it is assumed that $ A $ is canonically imbedded in $ A ^ {\prime\prime} $).

References

[1] M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)
[2] J. Bognár, "Certain relations among the non-negativity properties of operators on spaces with an indefinite metric II" Stud. Scient. Math. Hung. , 1 : 1–2 (1966) pp. 97–102 (In Russian)
[3] J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)

V.S. Shul'man

Comments

References

[a1] I.C. [I.Ts. Gokhberg] Gohberg, M.G. Krein, "Introduction to the theory of linear nonselfadjoint operators" , Transl. Math. Monogr. , 18 , Amer. Math. Soc. (1969) (Translated from Russian)
How to Cite This Entry:
Polar decomposition. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polar_decomposition&oldid=48228
This article was adapted from an original article by A.L. Onishchik, V.S. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article