Namespaces
Variants
Actions

Difference between revisions of "Nuclear norm"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 +
<!--
 +
n0678401.png
 +
$#A+1 = 107 n = 1
 +
$#C+1 = 107 : ~/encyclopedia/old_files/data/N067/N.0607840 Nuclear norm,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''trace norm''
 
''trace norm''
  
A norm on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n0678401.png" /> of nuclear operators (cf. [[Nuclear operator|Nuclear operator]]) mapping a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n0678402.png" /> into a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n0678403.png" />.
+
A norm on the space $  N ( X, Y) $
 +
of nuclear operators (cf. [[Nuclear operator|Nuclear operator]]) mapping a [[Banach space|Banach space]] $  X $
 +
into a Banach space $  Y $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n0678404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n0678405.png" /> be Banach spaces over the field of real or complex numbers, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n0678406.png" /> be the space of all continuous linear operators mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n0678407.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n0678408.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n0678409.png" /> be the linear subspace consisting of operators of finite rank (that is, with finite-dimensional range). The Banach dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784010.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784011.png" />, and the value of a functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784012.png" /> at a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784013.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784014.png" />.
+
Let $  X $
 +
and $  Y $
 +
be Banach spaces over the field of real or complex numbers, let $  L ( X, Y) $
 +
be the space of all continuous linear operators mapping $  X $
 +
into $  Y $,  
 +
and let $  F ( X, Y) $
 +
be the linear subspace consisting of operators of finite rank (that is, with finite-dimensional range). The Banach dual of $  X $
 +
is denoted by $  X  ^  \prime  $,  
 +
and the value of a functional $  x  ^  \prime  \in X  ^  \prime  $
 +
at a vector $  x \in X $
 +
by $  \langle  x, x  ^  \prime  \rangle $.
  
Every nuclear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784015.png" /> can be represented in the form
+
Every nuclear operator $  A \in N ( X, Y) $
 +
can be represented 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/n/n067/n067840/n06784016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
x  \mapsto  Ax  = \
 +
\sum _ {i = 1 } ^  \infty 
 +
\langle  x, x _ {i}  ^  \prime  \rangle y _ {i} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784018.png" /> are sequences in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784020.png" />, respectively, such that
+
where $  \{ x _ {i}  ^  \prime  \} $
 +
and $  \{ y _ {i} \} $
 +
are sequences in $  X  ^  \prime  $
 +
and $  Y $,  
 +
respectively, 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/n067840/n06784021.png" /></td> </tr></table>
+
$$
 +
\sum _ {i = 1 } ^  \infty 
 +
\| x _ {i}  ^  \prime  \|  \| y _ {i} \|  < \infty ;
 +
$$
  
 
such representations are called nuclear. The quantity
 
such representations are called nuclear. The quantity
  
<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/n067840/n06784022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\| A \| _ {1}  = \
 +
\inf \
 +
\sum _ {i = 1 } ^  \infty 
 +
\| x _ {i}  ^  \prime  \|  \| y _ {i} \| ,
 +
$$
  
where the infimum is taken over all possible nuclear representations of the form (1), is called the nuclear norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784023.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784024.png" /> with this norm is a Banach space that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784025.png" /> as a dense linear subspace. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784026.png" />, then the adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784027.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784028.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784029.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784030.png" /> denote the usual operator norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784031.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784032.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784033.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784035.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784036.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784037.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784039.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784040.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784041.png" />. Any operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784042.png" /> can be represented in the form
+
where the infimum is taken over all possible nuclear representations of the form (1), is called the nuclear norm of $  A $.  
 +
The space $  N ( X, Y) $
 +
with this norm is a Banach space that contains $  F ( X, Y) $
 +
as a dense linear subspace. If $  A \in N ( X, Y) $,  
 +
then the adjoint operator $  A  ^  \prime  $
 +
belongs to $  N ( Y  ^  \prime  , X  ^  \prime  ) $,  
 +
and $  \| A  ^  \prime  \| _ {1} \leq  \| A \| _ {1} $.  
 +
Let $  \|  \cdot  \| $
 +
denote the usual [[operator norm]] in $  L ( X, Y) $.  
 +
Then $  \| A \| \leq  \| A \| _ {1} $
 +
for all $  A \in N ( X, Y) $.  
 +
If $  A \in L ( Y, Z) $
 +
and $  B \in N ( X, Y) $,  
 +
then $  AB \in N ( X, Z) $,  
 +
and $  \| AB \| _ {1} \leq  \| A \|  \| B \| _ {1} $;  
 +
if $  A \in N ( Y, Z) $
 +
and $  B \in L ( X, Y) $,  
 +
then $  AB \in N ( X, Z) $,  
 +
and $  \| AB \| _ {1} \leq  \| A \| _ {1}  \| B \| $.  
 +
Any operator $  F \in F ( X, Y) $
 +
can be represented 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/n/n067/n067840/n06784043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
x  \mapsto  Fx  = \
 +
\sum _ {i = 1 } ^ { n }
 +
\langle  x, x _ {i}  ^  \prime  \rangle y _ {i} .
 +
$$
  
 
The quantity
 
The quantity
  
<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/n067840/n06784044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
\| F \| _ {1}  ^ {0}  = \
 +
\inf \
 +
\sum _ {i = 1 } ^ { n }
 +
\| x _ {i}  ^  \prime  \|  \| y _ {i} \| ,
 +
$$
  
where the infimum is taken over all possible finite representations of the form (3), is called the finite nuclear norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784045.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784046.png" /> can be identified with the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784047.png" />. Here, to an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784048.png" /> of the form (3) there corresponds the element
+
where the infimum is taken over all possible finite representations of the form (3), is called the finite nuclear norm of $  F $.  
 +
The space $  F ( X, Y) $
 +
can be identified with the tensor product $  X  ^  \prime  \otimes Y $.  
 +
Here, to an operator $  F $
 +
of the form (3) there corresponds the element
  
<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/n067840/n06784049.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
= \
 +
\sum _ {i = 1 } ^ { n }
 +
x _ {i}  ^  \prime  \otimes y _ {i}  \in \
 +
X  ^  \prime  \otimes Y,
 +
$$
  
 
and the finite nuclear norm (4) goes into the norm
 
and the finite nuclear norm (4) goes into the norm
  
<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/n067840/n06784050.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$ \tag{6 }
 +
\| u \|  = \
 +
\inf \
 +
\sum _ {i = 1 } ^ { n }
 +
\| x _ {i}  ^  \prime  \|  \| y _ {i} \| ,
 +
$$
  
where the infimum is taken over all finite representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784051.png" /> in the form (5). This norm is called the tensor (or cross) product of the norms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784052.png" /> and in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784053.png" />. The completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784054.png" /> with respect to the norm (6) is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784055.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784056.png" />, under which the element (5) is mapped to the operator (3), can be extended to a continuous linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784057.png" />. The range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784058.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784059.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784060.png" /> establishes a one-to-one correspondence between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784062.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784063.png" /> coincides with the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784064.png" /> with respect to the norm (4); in this case the restriction of the nuclear norm to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784065.png" /> is the same as the finite nuclear norm. But, in general, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784066.png" /> may have a non-trivial kernel, so that the nuclear norm is a quotient of the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784067.png" /> (see [[Nuclear operator|Nuclear operator]]).
+
where the infimum is taken over all finite representations of $  u $
 +
in the form (5). This norm is called the tensor (or cross) product of the norms in $  Y $
 +
and in $  X  ^  \prime  $.  
 +
The completion of $  X  ^  \prime  \otimes Y $
 +
with respect to the norm (6) is denoted by $  X  ^  \prime  \widehat \otimes  Y $.  
 +
The mapping $  X  ^  \prime  \otimes Y \rightarrow L ( X, Y) $,  
 +
under which the element (5) is mapped to the operator (3), can be extended to a continuous linear operator $  \Gamma : X  ^  \prime  \widehat \otimes  Y \rightarrow L ( X, Y) $.  
 +
The range of $  \Gamma $
 +
is $  N ( X, Y) $.  
 +
If $  \Gamma $
 +
establishes a one-to-one correspondence between $  X  ^  \prime  \widehat \otimes  Y $
 +
and $  N ( X, Y) $,  
 +
then $  N ( X, Y) $
 +
coincides with the closure of $  F ( X, Y) $
 +
with respect to the norm (4); in this case the restriction of the nuclear norm to $  F ( X, Y) $
 +
is the same as the finite nuclear norm. But, in general, $  \Gamma $
 +
may have a non-trivial kernel, so that the nuclear norm is a quotient of the norm in $  X  ^  \prime  \widehat \otimes  Y $(
 +
see [[Nuclear operator|Nuclear operator]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784068.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784069.png" /> is a separable Hilbert space, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784070.png" /> be the algebra of bounded operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784071.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784072.png" /> be the ideal of nuclear operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784073.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784074.png" /> is one-to-one, for operators of finite rank the nuclear norm coincides with the finite nuclear norm, and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784075.png" /> has a trace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784076.png" /> (see [[Nuclear operator|Nuclear operator]]). The nuclear norm of an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784077.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784078.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784079.png" /> is the adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784080.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784081.png" />. The nuclear norm is connected with the [[Hilbert–Schmidt norm|Hilbert–Schmidt norm]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784082.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784083.png" />. The general form of a continuous linear functional on the Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784084.png" /> is given by
+
Let $  X = Y = H $,  
 +
where $  H $
 +
is a separable Hilbert space, let $  L ( H) = L ( H, H) $
 +
be the algebra of bounded operators on $  H $,  
 +
and let $  L _ {1} ( H) = N ( H, H) $
 +
be the ideal of nuclear operators in $  L ( H) $.  
 +
In this case $  \Gamma $
 +
is one-to-one, for operators of finite rank the nuclear norm coincides with the finite nuclear norm, and each $  A \in L _ {1} ( H) $
 +
has a trace $  \mathop{\rm tr}  A $(
 +
see [[Nuclear operator|Nuclear operator]]). The nuclear norm of an operator $  A \in L _ {1} ( H) $
 +
coincides with $  \mathop{\rm tr}  [( A  ^ {*} A)  ^ {1/2} ] $,  
 +
where $  A  ^ {*} $
 +
is the adjoint of $  A $
 +
in $  H $.  
 +
The nuclear norm is connected with the [[Hilbert–Schmidt norm|Hilbert–Schmidt norm]] $  \|  \cdot  \| _ {2} $
 +
by $  \| A \| _ {2} \leq  \| A \| _ {1} $.  
 +
The general form of a continuous linear functional on the Banach space $  L _ {1} ( H) $
 +
is given 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/n/n067/n067840/n06784085.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
$$ \tag{7 }
 +
A  \rightarrow  \mathop{\rm tr}  AB,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784086.png" /> is an arbitrary operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784087.png" />, and the norm of the functional (7) coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784088.png" />. Consequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784089.png" /> is isometric to the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784090.png" />. Formula (7) also gives the general form of a linear functional on the closed subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784091.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784092.png" /> that consists of all completely-continuous (compact) operators; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784093.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784094.png" /> ranges over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784095.png" />. In this case the norm of the functional (7) coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784096.png" />, that is, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784097.png" /> of nuclear operators with the nuclear norm is isometric to the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784098.png" /> in the usual operator norm. These results have non-trivial generalizations to the case of operators on Banach spaces.
+
where $  B $
 +
is an arbitrary operator from $  L ( H) $,  
 +
and the norm of the functional (7) coincides with $  \| B \| $.  
 +
Consequently, $  L ( H) $
 +
is isometric to the dual of $  L _ {1} ( H) $.  
 +
Formula (7) also gives the general form of a linear functional on the closed subspace $  L _  \infty  ( H) $
 +
of $  L ( H) $
 +
that consists of all completely-continuous (compact) operators; here $  A \in L _  \infty  ( H) $
 +
and $  B $
 +
ranges over $  L _ {1} ( H) $.  
 +
In this case the norm of the functional (7) coincides with $  \| B \| _ {1} $,  
 +
that is, the space $  L _ {1} ( H) $
 +
of nuclear operators with the nuclear norm is isometric to the dual of $  L _  \infty  ( H) $
 +
in the usual operator norm. These results have non-trivial generalizations to the case of operators on Banach spaces.
  
Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n06784099.png" /> be the space of summable sequences. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n067840100.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n067840101.png" /> if and only if there is an infinite matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n067840102.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n067840103.png" /> sends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n067840104.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n067840105.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n067840106.png" />. In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n067840107.png" />.
+
Example. Let $  X = Y = l _ {1} $
 +
be the space of summable sequences. An operator $  A \in L ( l _ {1} , l _ {1} ) $
 +
is contained in $  N ( l _ {1} , l _ {1} ) $
 +
if and only if there is an infinite matrix $  ( \sigma _ {ik} ) $
 +
such that $  A $
 +
sends $  \{ \xi _ {k} \} \in l _ {1} $
 +
to $  \{ \eta _ {i} \} = \{ \sum _ {k = 1 }  ^  \infty  \sigma _ {ik} \xi _ {k} \} \in l _ {1} $,  
 +
and $  \sum _ {i = 1 }  ^  \infty  \sup _ {k}  | \sigma _ {ik} | < \infty $.  
 +
In this case, $  \| A \| _ {1} = \sum _ {i = 1 }  ^  \infty  \sup _ {k}  | \sigma _ {ik} | $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,  "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc.  (1955)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Pietsch,  "Operator ideals" , North-Holland  (1980)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Pietsch,  "Nuclear locally convex spaces" , Springer  (1972)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</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><TR><TD valign="top">[5]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. Maurin,  "Methods of Hilbert spaces" , PWN  (1967)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  M.M. Day,  "Normed linear spaces" , Springer  (1958)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,  "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc.  (1955)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Pietsch,  "Operator ideals" , North-Holland  (1980)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Pietsch,  "Nuclear locally convex spaces" , Springer  (1972)  (Translated from German)</TD></TR><TR><TD valign="top">[4]</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><TR><TD valign="top">[5]</TD> <TD valign="top">  I.M. Gel'fand,  N.Ya. Vilenkin,  "Generalized functions. Applications of harmonic analysis" , '''4''' , Acad. Press  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  K. Maurin,  "Methods of Hilbert spaces" , PWN  (1967)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  M.M. Day,  "Normed linear spaces" , Springer  (1958)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Pietsch,  "Eigenvalues and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n067840108.png" />-numbers" , Cambridge Univ. Press  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Grothendieck,  "Résumé de la théorie métrique des produits tensoriels topologiques"  ''Bol. Soc. Mat. São Paulo'' , '''8'''  (1956)  pp. 1–79</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Jarchow,  "Locally convex spaces" , Teubner  (1981)  (Translated from German)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Pietsch,  "Eigenvalues and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067840/n067840108.png" />-numbers" , Cambridge Univ. Press  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Grothendieck,  "Résumé de la théorie métrique des produits tensoriels topologiques"  ''Bol. Soc. Mat. São Paulo'' , '''8'''  (1956)  pp. 1–79</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Jarchow,  "Locally convex spaces" , Teubner  (1981)  (Translated from German)</TD></TR></table>

Latest revision as of 08:03, 6 June 2020


trace norm

A norm on the space $ N ( X, Y) $ of nuclear operators (cf. Nuclear operator) mapping a Banach space $ X $ into a Banach space $ Y $.

Let $ X $ and $ Y $ be Banach spaces over the field of real or complex numbers, let $ L ( X, Y) $ be the space of all continuous linear operators mapping $ X $ into $ Y $, and let $ F ( X, Y) $ be the linear subspace consisting of operators of finite rank (that is, with finite-dimensional range). The Banach dual of $ X $ is denoted by $ X ^ \prime $, and the value of a functional $ x ^ \prime \in X ^ \prime $ at a vector $ x \in X $ by $ \langle x, x ^ \prime \rangle $.

Every nuclear operator $ A \in N ( X, Y) $ can be represented in the form

$$ \tag{1 } x \mapsto Ax = \ \sum _ {i = 1 } ^ \infty \langle x, x _ {i} ^ \prime \rangle y _ {i} , $$

where $ \{ x _ {i} ^ \prime \} $ and $ \{ y _ {i} \} $ are sequences in $ X ^ \prime $ and $ Y $, respectively, such that

$$ \sum _ {i = 1 } ^ \infty \| x _ {i} ^ \prime \| \| y _ {i} \| < \infty ; $$

such representations are called nuclear. The quantity

$$ \tag{2 } \| A \| _ {1} = \ \inf \ \sum _ {i = 1 } ^ \infty \| x _ {i} ^ \prime \| \| y _ {i} \| , $$

where the infimum is taken over all possible nuclear representations of the form (1), is called the nuclear norm of $ A $. The space $ N ( X, Y) $ with this norm is a Banach space that contains $ F ( X, Y) $ as a dense linear subspace. If $ A \in N ( X, Y) $, then the adjoint operator $ A ^ \prime $ belongs to $ N ( Y ^ \prime , X ^ \prime ) $, and $ \| A ^ \prime \| _ {1} \leq \| A \| _ {1} $. Let $ \| \cdot \| $ denote the usual operator norm in $ L ( X, Y) $. Then $ \| A \| \leq \| A \| _ {1} $ for all $ A \in N ( X, Y) $. If $ A \in L ( Y, Z) $ and $ B \in N ( X, Y) $, then $ AB \in N ( X, Z) $, and $ \| AB \| _ {1} \leq \| A \| \| B \| _ {1} $; if $ A \in N ( Y, Z) $ and $ B \in L ( X, Y) $, then $ AB \in N ( X, Z) $, and $ \| AB \| _ {1} \leq \| A \| _ {1} \| B \| $. Any operator $ F \in F ( X, Y) $ can be represented in the form

$$ \tag{3 } x \mapsto Fx = \ \sum _ {i = 1 } ^ { n } \langle x, x _ {i} ^ \prime \rangle y _ {i} . $$

The quantity

$$ \tag{4 } \| F \| _ {1} ^ {0} = \ \inf \ \sum _ {i = 1 } ^ { n } \| x _ {i} ^ \prime \| \| y _ {i} \| , $$

where the infimum is taken over all possible finite representations of the form (3), is called the finite nuclear norm of $ F $. The space $ F ( X, Y) $ can be identified with the tensor product $ X ^ \prime \otimes Y $. Here, to an operator $ F $ of the form (3) there corresponds the element

$$ \tag{5 } u = \ \sum _ {i = 1 } ^ { n } x _ {i} ^ \prime \otimes y _ {i} \in \ X ^ \prime \otimes Y, $$

and the finite nuclear norm (4) goes into the norm

$$ \tag{6 } \| u \| = \ \inf \ \sum _ {i = 1 } ^ { n } \| x _ {i} ^ \prime \| \| y _ {i} \| , $$

where the infimum is taken over all finite representations of $ u $ in the form (5). This norm is called the tensor (or cross) product of the norms in $ Y $ and in $ X ^ \prime $. The completion of $ X ^ \prime \otimes Y $ with respect to the norm (6) is denoted by $ X ^ \prime \widehat \otimes Y $. The mapping $ X ^ \prime \otimes Y \rightarrow L ( X, Y) $, under which the element (5) is mapped to the operator (3), can be extended to a continuous linear operator $ \Gamma : X ^ \prime \widehat \otimes Y \rightarrow L ( X, Y) $. The range of $ \Gamma $ is $ N ( X, Y) $. If $ \Gamma $ establishes a one-to-one correspondence between $ X ^ \prime \widehat \otimes Y $ and $ N ( X, Y) $, then $ N ( X, Y) $ coincides with the closure of $ F ( X, Y) $ with respect to the norm (4); in this case the restriction of the nuclear norm to $ F ( X, Y) $ is the same as the finite nuclear norm. But, in general, $ \Gamma $ may have a non-trivial kernel, so that the nuclear norm is a quotient of the norm in $ X ^ \prime \widehat \otimes Y $( see Nuclear operator).

Let $ X = Y = H $, where $ H $ is a separable Hilbert space, let $ L ( H) = L ( H, H) $ be the algebra of bounded operators on $ H $, and let $ L _ {1} ( H) = N ( H, H) $ be the ideal of nuclear operators in $ L ( H) $. In this case $ \Gamma $ is one-to-one, for operators of finite rank the nuclear norm coincides with the finite nuclear norm, and each $ A \in L _ {1} ( H) $ has a trace $ \mathop{\rm tr} A $( see Nuclear operator). The nuclear norm of an operator $ A \in L _ {1} ( H) $ coincides with $ \mathop{\rm tr} [( A ^ {*} A) ^ {1/2} ] $, where $ A ^ {*} $ is the adjoint of $ A $ in $ H $. The nuclear norm is connected with the Hilbert–Schmidt norm $ \| \cdot \| _ {2} $ by $ \| A \| _ {2} \leq \| A \| _ {1} $. The general form of a continuous linear functional on the Banach space $ L _ {1} ( H) $ is given by

$$ \tag{7 } A \rightarrow \mathop{\rm tr} AB, $$

where $ B $ is an arbitrary operator from $ L ( H) $, and the norm of the functional (7) coincides with $ \| B \| $. Consequently, $ L ( H) $ is isometric to the dual of $ L _ {1} ( H) $. Formula (7) also gives the general form of a linear functional on the closed subspace $ L _ \infty ( H) $ of $ L ( H) $ that consists of all completely-continuous (compact) operators; here $ A \in L _ \infty ( H) $ and $ B $ ranges over $ L _ {1} ( H) $. In this case the norm of the functional (7) coincides with $ \| B \| _ {1} $, that is, the space $ L _ {1} ( H) $ of nuclear operators with the nuclear norm is isometric to the dual of $ L _ \infty ( H) $ in the usual operator norm. These results have non-trivial generalizations to the case of operators on Banach spaces.

Example. Let $ X = Y = l _ {1} $ be the space of summable sequences. An operator $ A \in L ( l _ {1} , l _ {1} ) $ is contained in $ N ( l _ {1} , l _ {1} ) $ if and only if there is an infinite matrix $ ( \sigma _ {ik} ) $ such that $ A $ sends $ \{ \xi _ {k} \} \in l _ {1} $ to $ \{ \eta _ {i} \} = \{ \sum _ {k = 1 } ^ \infty \sigma _ {ik} \xi _ {k} \} \in l _ {1} $, and $ \sum _ {i = 1 } ^ \infty \sup _ {k} | \sigma _ {ik} | < \infty $. In this case, $ \| A \| _ {1} = \sum _ {i = 1 } ^ \infty \sup _ {k} | \sigma _ {ik} | $.

References

[1] A. Grothendieck, "Produits tensoriels topologiques et espaces nucléaires" , Amer. Math. Soc. (1955)
[2] A. Pietsch, "Operator ideals" , North-Holland (1980)
[3] A. Pietsch, "Nuclear locally convex spaces" , Springer (1972) (Translated from German)
[4] 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)
[5] I.M. Gel'fand, N.Ya. Vilenkin, "Generalized functions. Applications of harmonic analysis" , 4 , Acad. Press (1968) (Translated from Russian)
[6] K. Maurin, "Methods of Hilbert spaces" , PWN (1967)
[7] M.M. Day, "Normed linear spaces" , Springer (1958)

Comments

References

[a1] A. Pietsch, "Eigenvalues and -numbers" , Cambridge Univ. Press (1987)
[a2] A. Grothendieck, "Résumé de la théorie métrique des produits tensoriels topologiques" Bol. Soc. Mat. São Paulo , 8 (1956) pp. 1–79
[a3] H. Jarchow, "Locally convex spaces" , Teubner (1981) (Translated from German)
How to Cite This Entry:
Nuclear norm. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nuclear_norm&oldid=19242
This article was adapted from an original article by G.L. Litvinov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article