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''on kernel operators''
 
''on kernel operators''
  
The theory of kernel operators (cf. also [[Kernel of an integral operator|Kernel of an integral operator]]) was essentially influenced by N. Dunford and B.J. Pettis [[#References|[a2]]] around 1940. Other important results were obtained at about the same time by L.V. Kantorovich and B.Z. Vulikh. Among other results, it was shown that every bounded [[Linear operator|linear operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b1205301.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b1205302.png" />, is a kernel operator in the sense that there exists a measurable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b1205303.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b1205304.png" /> for almost all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b1205305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b1205306.png" />.
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The theory of kernel operators (cf. also [[Kernel of an integral operator|Kernel of an integral operator]]) was essentially influenced by N. Dunford and B.J. Pettis [[#References|[a2]]] around 1940. Other important results were obtained at about the same time by L.V. Kantorovich and B.Z. Vulikh. Among other results, it was shown that every bounded [[Linear operator|linear operator]] $T : L ^ { 1 } ( \mu ) \rightarrow L ^ { p } ( \nu )$, $1 &lt; p \leq \infty$, is a kernel operator in the sense that there exists a measurable function $K$ such that $K ( ., s ) \in L ^ { 1 } ( \mu )$ for almost all $s$ and $(\; f \mapsto \int K ( t , \cdot ) f ( t ) d \mu ( t ) = T f ) \in L ^ { p } ( \nu )$.
  
 
This result is known as Dunford's theorem. In the decades following these first results, kernel operators were intensively studied. While the first results were mainly concerned with a single kernel operator or were in the spirit of Dunford's theorem, the investigation of the structure of the space of all kernel operators in the space of all regular operators began in the 1960s, following the study of Banach function spaces (cf. also [[Banach function space|Banach function space]]), or even ideal spaces in the space of measurable functions.
 
This result is known as Dunford's theorem. In the decades following these first results, kernel operators were intensively studied. While the first results were mainly concerned with a single kernel operator or were in the spirit of Dunford's theorem, the investigation of the structure of the space of all kernel operators in the space of all regular operators began in the 1960s, following the study of Banach function spaces (cf. also [[Banach function space|Banach function space]]), or even ideal spaces in the space of measurable functions.
  
Under very general assumptions it was shown that the kernel operators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b1205307.png" /> form a band in the space of all regular operators, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b1205308.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b1205309.png" /> are ideals in the space of measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053010.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053011.png" />, respectively. At that time many properties of kernel operators were known. It was A.V. Bukhvalov who gave [[#References|[a1]]] a simple characterization of kernel operators, as follows.
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Under very general assumptions it was shown that the kernel operators of $T : L \rightarrow M$ form a band in the space of all regular operators, where $L$ and $M$ are ideals in the space of measurable functions $M ( \mu )$ or $M ( \nu )$, respectively. At that time many properties of kernel operators were known. It was A.V. Bukhvalov who gave [[#References|[a1]]] a simple characterization of kernel operators, as follows.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053013.png" /> be measure spaces (cf. also [[Measure space|Measure space]]), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053015.png" /> be ideals such that the support of the Köthe dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053016.png" /> is all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053017.png" />. Then for every linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053018.png" /> the following conditions are equivalent:
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Let $( \Omega , A , \mu )$, $( \Omega _ { 1 } , A _ { 1 } , \nu )$ be measure spaces (cf. also [[Measure space|Measure space]]), let $L \subset M ( \mu )$ and $M \subset M ( \nu )$ be ideals such that the support of the Köthe dual $L ^ { \times }$ is all of $\Omega$. Then for every linear operator $T : L \rightarrow M$ the following conditions are equivalent:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053019.png" /> is a kernel operator.
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i) $T$ is a kernel operator.
  
ii) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053020.png" /> is an order-bounded sequence which is star convergent, then the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053021.png" /> is convergent almost everywhere. Here, a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053022.png" /> said to be star convergent to some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053023.png" /> if every subsequence of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053024.png" /> contains a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053025.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053026.png" /> almost everywhere as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053027.png" />. Consequently, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053028.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053029.png" />, if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053030.png" /> in the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053031.png" /> on every subset of finite measure.
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ii) If $( f _ { n } ) _ { n = 1 } ^ { \infty } \subset L _ { + }$ is an order-bounded sequence which is star convergent, then the sequence $( T f _ { n } ) _ { n = 1 } ^ { \infty } \subset M$ is convergent almost everywhere. Here, a sequence $( f _ { n } ) _ { n = 1 } ^ { \infty } $ said to be star convergent to some $f$ if every subsequence of the sequence $( f _ { n } ) _ { n = 1 } ^ { \infty } $ contains a subsequence $( h _ { n } ) _ { n = 1 } ^ { \infty } 1$ such that $h _ { n} \rightarrow f$ almost everywhere as $n \rightarrow \infty$. Consequently, $f _ { n } \rightarrow ^ { * } f$, as $n \rightarrow \infty$, if and only if $f _ { n } \rightarrow f$ in the measure $\mu$ on every subset of finite measure.
  
While the proof of i)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053032.png" />ii) is a simple consequence of the Lebesgue convergence theorem (cf. also [[Lebesgue theorem|Lebesgue theorem]]), the proof of ii)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120530/b12053033.png" />i) requires many results concerning the structure of the space of kernel operators in the space of all regular operators. A simplified version of the proof is due to A.R. Schep [[#References|[a4]]].
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While the proof of i)$\Rightarrow$ii) is a simple consequence of the Lebesgue convergence theorem (cf. also [[Lebesgue theorem|Lebesgue theorem]]), the proof of ii)$\Rightarrow$i) requires many results concerning the structure of the space of kernel operators in the space of all regular operators. A simplified version of the proof is due to A.R. Schep [[#References|[a4]]].
  
 
Bukhvalov's theorem is a powerful tool in the study of operators between Banach function spaces. In particular, his characterization of kernel operators leads to simple proofs of many classical results, such as Dunford's theorem and generalizations of it. For more information, see [[#References|[a3]]], Sect. 3.3, or [[#References|[a5]]].
 
Bukhvalov's theorem is a powerful tool in the study of operators between Banach function spaces. In particular, his characterization of kernel operators leads to simple proofs of many classical results, such as Dunford's theorem and generalizations of it. For more information, see [[#References|[a3]]], Sect. 3.3, or [[#References|[a5]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Bukhvalov,  "Integral representations of linear operators"  ''J. Soviet Math.'' , '''8'''  (1978)  pp. 129–137</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Dunford,  J.B. Pettis,  "Linear operators on summable functions"  ''Trans. Amer. Math. Soc.'' , '''47'''  (1940)  pp. 323–392</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Meyer-Nieberg,  "Banach lattices" , Springer  (1991)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.R. Schep,  "Kernel operators"  ''PhD Thesis Univ. Leiden''  (1977)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.C. Zaanen,  "Riesz spaces" , '''II''' , North-Holland  (1983)</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  A.V. Bukhvalov,  "Integral representations of linear operators"  ''J. Soviet Math.'' , '''8'''  (1978)  pp. 129–137</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  N. Dunford,  J.B. Pettis,  "Linear operators on summable functions"  ''Trans. Amer. Math. Soc.'' , '''47'''  (1940)  pp. 323–392</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  P. Meyer-Nieberg,  "Banach lattices" , Springer  (1991)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  A.R. Schep,  "Kernel operators"  ''PhD Thesis Univ. Leiden''  (1977)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A.C. Zaanen,  "Riesz spaces" , '''II''' , North-Holland  (1983)</td></tr></table>

Latest revision as of 15:30, 1 July 2020

on kernel operators

The theory of kernel operators (cf. also Kernel of an integral operator) was essentially influenced by N. Dunford and B.J. Pettis [a2] around 1940. Other important results were obtained at about the same time by L.V. Kantorovich and B.Z. Vulikh. Among other results, it was shown that every bounded linear operator $T : L ^ { 1 } ( \mu ) \rightarrow L ^ { p } ( \nu )$, $1 < p \leq \infty$, is a kernel operator in the sense that there exists a measurable function $K$ such that $K ( ., s ) \in L ^ { 1 } ( \mu )$ for almost all $s$ and $(\; f \mapsto \int K ( t , \cdot ) f ( t ) d \mu ( t ) = T f ) \in L ^ { p } ( \nu )$.

This result is known as Dunford's theorem. In the decades following these first results, kernel operators were intensively studied. While the first results were mainly concerned with a single kernel operator or were in the spirit of Dunford's theorem, the investigation of the structure of the space of all kernel operators in the space of all regular operators began in the 1960s, following the study of Banach function spaces (cf. also Banach function space), or even ideal spaces in the space of measurable functions.

Under very general assumptions it was shown that the kernel operators of $T : L \rightarrow M$ form a band in the space of all regular operators, where $L$ and $M$ are ideals in the space of measurable functions $M ( \mu )$ or $M ( \nu )$, respectively. At that time many properties of kernel operators were known. It was A.V. Bukhvalov who gave [a1] a simple characterization of kernel operators, as follows.

Let $( \Omega , A , \mu )$, $( \Omega _ { 1 } , A _ { 1 } , \nu )$ be measure spaces (cf. also Measure space), let $L \subset M ( \mu )$ and $M \subset M ( \nu )$ be ideals such that the support of the Köthe dual $L ^ { \times }$ is all of $\Omega$. Then for every linear operator $T : L \rightarrow M$ the following conditions are equivalent:

i) $T$ is a kernel operator.

ii) If $( f _ { n } ) _ { n = 1 } ^ { \infty } \subset L _ { + }$ is an order-bounded sequence which is star convergent, then the sequence $( T f _ { n } ) _ { n = 1 } ^ { \infty } \subset M$ is convergent almost everywhere. Here, a sequence $( f _ { n } ) _ { n = 1 } ^ { \infty } $ said to be star convergent to some $f$ if every subsequence of the sequence $( f _ { n } ) _ { n = 1 } ^ { \infty } $ contains a subsequence $( h _ { n } ) _ { n = 1 } ^ { \infty } 1$ such that $h _ { n} \rightarrow f$ almost everywhere as $n \rightarrow \infty$. Consequently, $f _ { n } \rightarrow ^ { * } f$, as $n \rightarrow \infty$, if and only if $f _ { n } \rightarrow f$ in the measure $\mu$ on every subset of finite measure.

While the proof of i)$\Rightarrow$ii) is a simple consequence of the Lebesgue convergence theorem (cf. also Lebesgue theorem), the proof of ii)$\Rightarrow$i) requires many results concerning the structure of the space of kernel operators in the space of all regular operators. A simplified version of the proof is due to A.R. Schep [a4].

Bukhvalov's theorem is a powerful tool in the study of operators between Banach function spaces. In particular, his characterization of kernel operators leads to simple proofs of many classical results, such as Dunford's theorem and generalizations of it. For more information, see [a3], Sect. 3.3, or [a5].

References

[a1] A.V. Bukhvalov, "Integral representations of linear operators" J. Soviet Math. , 8 (1978) pp. 129–137
[a2] N. Dunford, J.B. Pettis, "Linear operators on summable functions" Trans. Amer. Math. Soc. , 47 (1940) pp. 323–392
[a3] P. Meyer-Nieberg, "Banach lattices" , Springer (1991)
[a4] A.R. Schep, "Kernel operators" PhD Thesis Univ. Leiden (1977)
[a5] A.C. Zaanen, "Riesz spaces" , II , North-Holland (1983)
How to Cite This Entry:
Bukhvalov theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bukhvalov_theorem&oldid=17381
This article was adapted from an original article by Peter Meyer-Nieberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article