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Difference between revisions of "Integral representations of linear operators"

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m (fixing spaces)
 
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Let  $  ( X, \Lambda, \mu ) $
 
Let  $  ( X, \Lambda, \mu ) $
 
and  $  ( Y, \Sigma, \nu ) $
 
and  $  ( Y, \Sigma, \nu ) $
be  $  \sigma $-
+
be  $  \sigma $-finite measure spaces (cf. [[Measure space|Measure space]]) and let  $  L _ {0} ( X, \mu ) $
finite measure spaces (cf. [[Measure space|Measure space]]) and let  $  L _ {0} ( X, \mu ) $
 
 
and  $  L _ {0} ( Y, \nu ) $
 
and  $  L _ {0} ( Y, \nu ) $
be the spaces of the complex-valued  $  \mu $-
+
be the spaces of the complex-valued  $  \mu $-measurable functions on  $  X $
measurable functions on  $  X $
+
and the complex-valued  $  \nu $-measurable functions on  $  Y $,  
and the complex-valued  $  \nu $-
 
measurable functions on  $  Y $,  
 
 
respectively. A linear subspace  $  E = E ( X, \mu ) $
 
respectively. A linear subspace  $  E = E ( X, \mu ) $
 
of  $  L _ {0} ( X, \mu ) $
 
of  $  L _ {0} ( X, \mu ) $
Line 26: Line 23:
 
$  g \in E $
 
$  g \in E $
 
and  $  | f | \leq  | g | $,  
 
and  $  | f | \leq  | g | $,  
$  \mu $-
+
$  \mu $-a.e., imply  $  f \in E $.  
a.e., imply  $  f \in E $.  
+
The classical  $  L _ {p} $-spaces ( $  1 \leq  p \leq  \infty $),  
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.
 
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.
  
Line 39: Line 34:
 
the linear space of all linear operators from  $  E $
 
the linear space of all linear operators from  $  E $
 
into  $  F $,  
 
into  $  F $,  
is called an integral operator, kernel operator, if there exists a  $  ( \mu \times \nu ) $-
+
is called an integral operator, kernel operator, if there exists a  $  ( \mu \times \nu ) $-measurable function  $  T = T ( x,y ) $,  
measurable function  $  T = T ( x,y ) $,  
 
 
$  ( x,y ) \in X \times Y $,  
 
$  ( x,y ) \in X \times Y $,  
 
such that for all  $  f \in E $
 
such that for all  $  f \in E $
and  $  \mu $-
+
and  $  \mu $-a.e. with respect to  $  x $,  
a.e. with respect to  $  x $,  
 
 
$  ( Tf ) ( x ) = \int _ {Y} {T ( x,y ) f ( y ) }  {d \nu ( y ) } $.
 
$  ( Tf ) ( x ) = \int _ {Y} {T ( x,y ) f ( y ) }  {d \nu ( y ) } $.
  
Line 55: Line 48:
 
$  T \in {\mathcal L} ( E,F ) $
 
$  T \in {\mathcal L} ( E,F ) $
 
is called a positive linear operator if for all  $  0 \leq  f \in E $
 
is called a positive linear operator if for all  $  0 \leq  f \in E $
one has  $  T f \geq  0 $(
+
one has  $  T f \geq  0 $ ($  \mu $-a.e.). An integral operator  $  T $
$  \mu $-
+
with kernel  $  T ( x,y ) $ ($  ( x,y ) \in X \times Y $)  
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 $,  
 
is positive if and only if  $  T ( x,y ) \geq  0 $,  
$  ( \mu \times \nu ) $-
+
$  ( \mu \times \nu ) $-a.e.;  $  T \in {\mathcal L} ( E,F ) $
a.e.;  $  T \in {\mathcal L} ( E,F ) $
 
 
is called regular if  $  T $
 
is called regular if  $  T $
 
maps order-bounded sets into order-bounded sets, i.e., for all  $  f \in E $
 
maps order-bounded sets into order-bounded sets, i.e., for all  $  f \in E $
Line 82: Line 71:
 
into  $  F $.  
 
into  $  F $.  
 
In that case, the kernel of  $  | T | $
 
In that case, the kernel of  $  | T | $
is given by the modulus  $  | {T ( x,y ) } | $(
+
is given by the modulus  $  | {T ( x,y ) } | $ ($  ( x,y ) \in X \times Y $)  
$  ( x,y ) \in X \times Y $)  
 
 
of the kernel of  $  T $.
 
of the kernel of  $  T $.
  
Line 89: Line 77:
  
 
Integral operators can be characterized via a continuity property:  $  T \in {\mathcal L} ( E,F ) $
 
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 $(
+
is a linear integral operator if and only if  $  0 \leq  f _ {n} \leq  f \in E $ ($  n = 1,2, \dots $)  
$  n = 1,2, \dots $)  
 
 
and  $  f _ {n} \rightarrow 0 $
 
and  $  f _ {n} \rightarrow 0 $
in  $  \nu $-
+
in  $  \nu $-measure as  $  n \rightarrow \infty $
measure as  $  n \rightarrow \infty $
+
imply  $  Tf _ {n} \rightarrow 0 $ ($  \mu $-a.e.) 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 $-
+
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]]].
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>

Latest revision as of 16:51, 19 February 2022


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=47381
This article was adapted from an original article by W. Luxemburg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article