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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100801.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100802.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100803.png" />-finite measure spaces (cf. [[Measure space|Measure space]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100805.png" /> be the spaces of the complex-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100806.png" />-measurable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100807.png" /> and the complex-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100808.png" />-measurable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i1100809.png" />, respectively. A linear subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008011.png" /> is called an ideal space, or a solid linear subspace, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008012.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008016.png" />-a.e., imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008017.png" />. The classical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008019.png" />-spaces (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008020.png" />), the Orlicz spaces and, more generally, Banach function spaces (cf. also [[Orlicz space|Orlicz space]]; [[Banach space|Banach space]]) are typical examples of normed ideal spaces.
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008022.png" /> are ideal spaces contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008024.png" />, respectively, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008025.png" />, the linear space of all linear operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008026.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008027.png" />, is called an integral operator, kernel operator, if there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008028.png" />-measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008030.png" />, such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008032.png" />-a.e. with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008034.png" />.
+
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Integral operators, also known as integral transforms, play an important role in analysis. It is a natural question to ask: Which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008035.png" /> are integral operators? J. von Neumann [[#References|[a5]]] was the first to show that for the ideal spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008036.png" /> the identity operator does not admit an integral representation. He proved, however, that a bounded self-adjoint linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008037.png" /> is unitarily equivalent (cf. also [[Unitarily-equivalent operators|Unitarily-equivalent operators]]) to an integral operator if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008038.png" /> is an element of the limit spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008039.png" />.
+
Let  $  ( X, \Lambda, \mu ) $
 +
and  $  ( Y, \Sigma, \nu ) $
 +
be  $  \sigma $-
 +
finite measure spaces (cf. [[Measure space|Measure space]]) and let  $  L _ {0} ( X, \mu ) $
 +
and  $  L _ {0} ( Y, \nu ) $
 +
be the spaces of the complex-valued  $  \mu $-
 +
measurable functions on  $  X $
 +
and the complex-valued  $  \nu $-
 +
measurable functions on  $  Y $,
 +
respectively. A linear subspace  $  E = E ( X, \mu ) $
 +
of  $  L _ {0} ( X, \mu ) $
 +
is called an ideal space, or a solid linear subspace, of  $  L _ {0} $
 +
if  $  f \in L _ {0} $,
 +
$  g \in E $
 +
and  $  | f | \leq  | g | $,
 +
$  \mu $-
 +
a.e., imply  $  f \in E $.  
 +
The classical  $  L _ {p} $-
 +
spaces ( $  1 \leq  p \leq  \infty $),
 +
the Orlicz spaces and, more generally, Banach function spaces (cf. also [[Orlicz space|Orlicz space]]; [[Banach space|Banach space]]) are typical examples of normed ideal spaces.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008040.png" /> is called a positive linear operator if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008041.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008042.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008043.png" />-a.e.). An integral operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008044.png" /> with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008045.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008046.png" />) is positive if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008048.png" />-a.e.; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008049.png" /> is called regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008050.png" /> maps order-bounded sets into order-bounded sets, i.e., for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008051.png" /> there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008052.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008053.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008054.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008055.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008056.png" /> is ordered bounded if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008057.png" /> can be written as the difference of two positive linear operators if and only if its modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008058.png" />, where for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008060.png" />, is a positive linear operator mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008061.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008062.png" />.
+
If  $  E $,
 +
$  F $
 +
are ideal spaces contained in  $  L _ {0} ( Y, \nu ) $
 +
and  $  L _ {0} ( X, \mu ) $,
 +
respectively, then  $  T \in {\mathcal L} ( E,F ) $,
 +
the linear space of all linear operators from  $  E $
 +
into  $  F $,  
 +
is called an integral operator, kernel operator, if there exists a $  ( \mu \times \nu ) $-
 +
measurable function  $  T = T ( x,y ) $,
 +
$  ( x,y ) \in X \times Y $,
 +
such that for all $  f \in E $
 +
and $  \mu $-
 +
a.e. with respect to  $  x $,  
 +
$  ( Tf ) ( x ) = \int _ {Y} {T ( x,y ) f ( y ) }  {d \nu ( y ) } $.
  
The following theorem holds: An integral operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008063.png" /> is regular if and only if its modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008064.png" /> is a positive linear operator mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008065.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008066.png" />. In that case, the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008067.png" /> is given by the modulus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008068.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008069.png" />) of the kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008070.png" />.
+
Integral operators, also known as integral transforms, play an important role in analysis. It is a natural question to ask: Which  $  T \in {\mathcal L} ( E,F ) $
 +
are integral operators? J. von Neumann [[#References|[a5]]] was the first to show that for the ideal spaces  $  E = F = L _ {2} ( [ 0,1 ] ) $
 +
the identity operator does not admit an integral representation. He proved, however, that a bounded self-adjoint linear operator  $  T \in {\mathcal L} ( L _ {2} , L _ {2} ) $
 +
is unitarily equivalent (cf. also [[Unitarily-equivalent operators|Unitarily-equivalent operators]]) to an integral operator if and only if 0 $
 +
is an element of the limit spectrum of  $  T $.
 +
 
 +
$  T \in {\mathcal L} ( E,F ) $
 +
is called a positive linear operator if for all  $  0 \leq  f \in E $
 +
one has  $  T f \geq  0 $(
 +
$  \mu $-
 +
a.e.). An integral operator  $  T $
 +
with kernel  $  T ( x,y ) $(
 +
$  ( x,y ) \in X \times Y $)
 +
is positive if and only if  $  T ( x,y ) \geq  0 $,
 +
$  ( \mu \times \nu ) $-
 +
a.e.;  $  T \in {\mathcal L} ( E,F ) $
 +
is called regular if  $  T $
 +
maps order-bounded sets into order-bounded sets, i.e., for all  $  f \in E $
 +
there exists a  $  g \in F $
 +
such that for all  $  h \in E $
 +
satisfying  $  | h | \leq  | f | $,
 +
one has  $  | {Th } | \leq  g $;
 +
$  T \in {\mathcal L} ( E,F ) $
 +
is ordered bounded if and only if  $  T $
 +
can be written as the difference of two positive linear operators if and only if its modulus  $  | T | $,
 +
where for all  $  0 \leq  f \in E $,
 +
$  | T | ( f ) = \sup  \{ {| {Tg } | } : {| g | \leq  f } \} $,
 +
is a positive linear operator mapping  $  E $
 +
into  $  F $.
 +
 
 +
The following theorem holds: An integral operator  $  T \in {\mathcal L} ( E,F ) $
 +
is regular if and only if its modulus  $  | T | $
 +
is a positive linear operator mapping  $  E $
 +
into  $  F $.  
 +
In that case, the kernel of  $  | T | $
 +
is given by the modulus $  | {T ( x,y ) } | $(
 +
( x,y ) \in X \times Y $)  
 +
of the kernel of $  T $.
  
 
An integral transform need not be regular, as is shown, for instance, by the [[Fourier transform|Fourier transform]] and the [[Hilbert transform|Hilbert transform]].
 
An integral transform need not be regular, as is shown, for instance, by the [[Fourier transform|Fourier transform]] and the [[Hilbert transform|Hilbert transform]].
  
Integral operators can be characterized via a continuity property: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008071.png" /> is a linear integral operator if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008072.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008073.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008074.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008075.png" />-measure as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008076.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008077.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008078.png" />-a.e.) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008079.png" />.
+
Integral operators can be characterized via a continuity property: $  T \in {\mathcal L} ( E,F ) $
 +
is a linear integral operator if and only if 0 \leq  f _ {n} \leq  f \in E $(
 +
$  n = 1,2, \dots $)  
 +
and $  f _ {n} \rightarrow 0 $
 +
in $  \nu $-
 +
measure as $  n \rightarrow \infty $
 +
imply $  Tf _ {n} \rightarrow 0 $(
 +
$  \mu $-
 +
a.e.) as $  n \rightarrow \infty $.
  
An earlier version of this theorem for bilinear forms is due to H. Nakano [[#References|[a4]]]. For regular linear operators defined on KB-spaces (cf. also [[K-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008080.png" />-space]]), the result appeared in a slightly different form in a paper by G.Ya. Lozonovskii [[#References|[a3]]]. The present version is due to A.V. Bukhvalov [[#References|[a1]]]. A pure measure-theoretic proof and related results were given by A. Schep [[#References|[a6]]]. For details and further results see [[#References|[a2]]].
+
An earlier version of this theorem for bilinear forms is due to H. Nakano [[#References|[a4]]]. For regular linear operators defined on KB-spaces (cf. also [[K-space| $  K $-
 +
space]]), the result appeared in a slightly different form in a paper by G.Ya. Lozonovskii [[#References|[a3]]]. The present version is due to A.V. Bukhvalov [[#References|[a1]]]. A pure measure-theoretic proof and related results were given by A. Schep [[#References|[a6]]]. For details and further results see [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Bukhvalov,  "A criterion for integral representability of linear operators"  ''Funktsional. Anal. i Prilozhen.'' , '''9''' :  1  (1975)  pp. 51  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  "Vector lattices and integral operators"  S.S. Kutateladze (ed.) , ''Mathematics and its Applications'' , '''358''' , Kluwer Acad. Publ.  (1996)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.Ya. Lozanovsky,  "On almost integral operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008081.png" />-spaces"  ''Vestnik Leningrad Gos. Univ.'' , '''7'''  (1966)  pp. 35–44  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Nakano,  "Product spaces of semi-ordered linear spaces"  ''J. Fac. Sci. Hokkaidô Univ. Ser. I'' , '''12''' :  3  (1953)  pp. 163–210</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. von Neumann,  "Charakterisierung des Spektrums eines Integraloperators" , ''Actualités Sc. et Industr.'' , '''229''' , Hermann  (1935)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.R. Schep,  "Kernel operators"  ''Proc. Kon. Nederl. Akad. Wetensch.'' , '''A 82'''  (1979)  pp. 39–53</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Bukhvalov,  "A criterion for integral representability of linear operators"  ''Funktsional. Anal. i Prilozhen.'' , '''9''' :  1  (1975)  pp. 51  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  "Vector lattices and integral operators"  S.S. Kutateladze (ed.) , ''Mathematics and its Applications'' , '''358''' , Kluwer Acad. Publ.  (1996)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.Ya. Lozanovsky,  "On almost integral operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i110/i110080/i11008081.png" />-spaces"  ''Vestnik Leningrad Gos. Univ.'' , '''7'''  (1966)  pp. 35–44  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Nakano,  "Product spaces of semi-ordered linear spaces"  ''J. Fac. Sci. Hokkaidô Univ. Ser. I'' , '''12''' :  3  (1953)  pp. 163–210</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. von Neumann,  "Charakterisierung des Spektrums eines Integraloperators" , ''Actualités Sc. et Industr.'' , '''229''' , Hermann  (1935)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.R. Schep,  "Kernel operators"  ''Proc. Kon. Nederl. Akad. Wetensch.'' , '''A 82'''  (1979)  pp. 39–53</TD></TR></table>

Revision as of 22:12, 5 June 2020


Let $ ( X, \Lambda, \mu ) $ and $ ( Y, \Sigma, \nu ) $ be $ \sigma $- finite measure spaces (cf. Measure space) and let $ L _ {0} ( X, \mu ) $ and $ L _ {0} ( Y, \nu ) $ be the spaces of the complex-valued $ \mu $- measurable functions on $ X $ and the complex-valued $ \nu $- measurable functions on $ Y $, respectively. A linear subspace $ E = E ( X, \mu ) $ of $ L _ {0} ( X, \mu ) $ is called an ideal space, or a solid linear subspace, of $ L _ {0} $ if $ f \in L _ {0} $, $ g \in E $ and $ | f | \leq | g | $, $ \mu $- a.e., imply $ f \in E $. The classical $ L _ {p} $- spaces ( $ 1 \leq p \leq \infty $), the Orlicz spaces and, more generally, Banach function spaces (cf. also Orlicz space; Banach space) are typical examples of normed ideal spaces.

If $ E $, $ F $ are ideal spaces contained in $ L _ {0} ( Y, \nu ) $ and $ L _ {0} ( X, \mu ) $, respectively, then $ T \in {\mathcal L} ( E,F ) $, the linear space of all linear operators from $ E $ into $ F $, is called an integral operator, kernel operator, if there exists a $ ( \mu \times \nu ) $- measurable function $ T = T ( x,y ) $, $ ( x,y ) \in X \times Y $, such that for all $ f \in E $ and $ \mu $- a.e. with respect to $ x $, $ ( Tf ) ( x ) = \int _ {Y} {T ( x,y ) f ( y ) } {d \nu ( y ) } $.

Integral operators, also known as integral transforms, play an important role in analysis. It is a natural question to ask: Which $ T \in {\mathcal L} ( E,F ) $ are integral operators? J. von Neumann [a5] was the first to show that for the ideal spaces $ E = F = L _ {2} ( [ 0,1 ] ) $ the identity operator does not admit an integral representation. He proved, however, that a bounded self-adjoint linear operator $ T \in {\mathcal L} ( L _ {2} , L _ {2} ) $ is unitarily equivalent (cf. also Unitarily-equivalent operators) to an integral operator if and only if $ 0 $ is an element of the limit spectrum of $ T $.

$ T \in {\mathcal L} ( E,F ) $ is called a positive linear operator if for all $ 0 \leq f \in E $ one has $ T f \geq 0 $( $ \mu $- a.e.). An integral operator $ T $ with kernel $ T ( x,y ) $( $ ( x,y ) \in X \times Y $) is positive if and only if $ T ( x,y ) \geq 0 $, $ ( \mu \times \nu ) $- a.e.; $ T \in {\mathcal L} ( E,F ) $ is called regular if $ T $ maps order-bounded sets into order-bounded sets, i.e., for all $ f \in E $ there exists a $ g \in F $ such that for all $ h \in E $ satisfying $ | h | \leq | f | $, one has $ | {Th } | \leq g $; $ T \in {\mathcal L} ( E,F ) $ is ordered bounded if and only if $ T $ can be written as the difference of two positive linear operators if and only if its modulus $ | T | $, where for all $ 0 \leq f \in E $, $ | T | ( f ) = \sup \{ {| {Tg } | } : {| g | \leq f } \} $, is a positive linear operator mapping $ E $ into $ F $.

The following theorem holds: An integral operator $ T \in {\mathcal L} ( E,F ) $ is regular if and only if its modulus $ | T | $ is a positive linear operator mapping $ E $ into $ F $. In that case, the kernel of $ | T | $ is given by the modulus $ | {T ( x,y ) } | $( $ ( x,y ) \in X \times Y $) of the kernel of $ T $.

An integral transform need not be regular, as is shown, for instance, by the Fourier transform and the Hilbert transform.

Integral operators can be characterized via a continuity property: $ T \in {\mathcal L} ( E,F ) $ is a linear integral operator if and only if $ 0 \leq f _ {n} \leq f \in E $( $ n = 1,2, \dots $) and $ f _ {n} \rightarrow 0 $ in $ \nu $- measure as $ n \rightarrow \infty $ imply $ Tf _ {n} \rightarrow 0 $( $ \mu $- a.e.) as $ n \rightarrow \infty $.

An earlier version of this theorem for bilinear forms is due to H. Nakano [a4]. For regular linear operators defined on KB-spaces (cf. also $ K $- space), the result appeared in a slightly different form in a paper by G.Ya. Lozonovskii [a3]. The present version is due to A.V. Bukhvalov [a1]. A pure measure-theoretic proof and related results were given by A. Schep [a6]. For details and further results see [a2].

References

[a1] A.V. Bukhvalov, "A criterion for integral representability of linear operators" Funktsional. Anal. i Prilozhen. , 9 : 1 (1975) pp. 51 (In Russian)
[a2] "Vector lattices and integral operators" S.S. Kutateladze (ed.) , Mathematics and its Applications , 358 , Kluwer Acad. Publ. (1996)
[a3] G.Ya. Lozanovsky, "On almost integral operators in -spaces" Vestnik Leningrad Gos. Univ. , 7 (1966) pp. 35–44 (In Russian)
[a4] H. Nakano, "Product spaces of semi-ordered linear spaces" J. Fac. Sci. Hokkaidô Univ. Ser. I , 12 : 3 (1953) pp. 163–210
[a5] J. von Neumann, "Charakterisierung des Spektrums eines Integraloperators" , Actualités Sc. et Industr. , 229 , Hermann (1935)
[a6] A.R. Schep, "Kernel operators" Proc. Kon. Nederl. Akad. Wetensch. , A 82 (1979) pp. 39–53
How to Cite This Entry:
Integral representations of linear operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_representations_of_linear_operators&oldid=19185
This article was adapted from an original article by W. Luxemburg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article