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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.

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 [[#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]].

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| $  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====
<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>