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A Carleman operator on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110090/c1100901.png" /> is an [[Integral operator|integral operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110090/c1100902.png" />, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110090/c1100903.png" /> a.e. for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110090/c1100904.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110090/c1100905.png" /> a.e. on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110090/c1100906.png" />. Self-adjoint Carleman operators have generalized eigenfunction expansions (cf. [[Eigen function|Eigen function]]; [[Series expansion|Series expansion]]), which can be used in the study of linear elliptic operators, see [[#References|[a1]]]. A general reference for Carleman operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110090/c1100907.png" />-spaces is [[#References|[a2]]]. The notion of a Carleman operator has been extended in many directions. By replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110090/c1100908.png" /> by an arbitrary Banach function space one obtains the so-called generalized Carleman operators (see [[#References|[a3]]]) and by considering Bochner integrals (cf. [[Bochner integral|Bochner integral]]) and abstract Banach spaces one is lead to the so-called Carleman and Korotkov operators on a [[Banach space|Banach space]] ([[#References|[a4]]]).
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A Carleman operator on the space  $  L _ {2} ( X, \mu ) $
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is an [[Integral operator|integral operator]] $  T $,  
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i.e., $  Tf ( x ) = \int {T ( x,y ) f ( y ) }  {d \mu ( y ) } $
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a.e. for $  f \in L _ {2} ( X, \mu ) $,  
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such that $  \| {T ( x, \cdot ) } \| _ {2} < \infty $
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a.e. on $  X $.  
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Self-adjoint Carleman operators have generalized eigenfunction expansions (cf. [[Eigen function|Eigen function]]; [[Series expansion|Series expansion]]), which can be used in the study of linear elliptic operators, see [[#References|[a1]]]. A general reference for Carleman operators on $  L _ {2} $-
 +
spaces is [[#References|[a2]]]. The notion of a Carleman operator has been extended in many directions. By replacing $  L _ {2} $
 +
by an arbitrary Banach function space one obtains the so-called generalized Carleman operators (see [[#References|[a3]]]) and by considering Bochner integrals (cf. [[Bochner integral|Bochner integral]]) and abstract Banach spaces one is lead to the so-called Carleman and Korotkov operators on a [[Banach space|Banach space]] ([[#References|[a4]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Maurin,  "Methods of Hilbert spaces" , PWN  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,  V.S. Sunder,  "Bounded integral operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110090/c1100909.png" />-spaces" , ''Ergebnisse der Mathematik und ihrer Grenzgebiete'' , '''96''' , Springer  (1978)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.R. Schep,  "Generalized Carleman operators"  ''Indagationes Mathematicae'' , '''42'''  (1980)  pp. 49–59</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Gretsky,  J.J. Uhl,  "Carleman and Korotkov operators on Banach spaces"  ''Acta Sci. Math'' , '''43'''  (1981)  pp. 111–119</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Maurin,  "Methods of Hilbert spaces" , PWN  (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,  V.S. Sunder,  "Bounded integral operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110090/c1100909.png" />-spaces" , ''Ergebnisse der Mathematik und ihrer Grenzgebiete'' , '''96''' , Springer  (1978)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.R. Schep,  "Generalized Carleman operators"  ''Indagationes Mathematicae'' , '''42'''  (1980)  pp. 49–59</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  N. Gretsky,  J.J. Uhl,  "Carleman and Korotkov operators on Banach spaces"  ''Acta Sci. Math'' , '''43'''  (1981)  pp. 111–119</TD></TR></table>

Latest revision as of 10:23, 2 June 2020


A Carleman operator on the space $ L _ {2} ( X, \mu ) $ is an integral operator $ T $, i.e., $ Tf ( x ) = \int {T ( x,y ) f ( y ) } {d \mu ( y ) } $ a.e. for $ f \in L _ {2} ( X, \mu ) $, such that $ \| {T ( x, \cdot ) } \| _ {2} < \infty $ a.e. on $ X $. Self-adjoint Carleman operators have generalized eigenfunction expansions (cf. Eigen function; Series expansion), which can be used in the study of linear elliptic operators, see [a1]. A general reference for Carleman operators on $ L _ {2} $- spaces is [a2]. The notion of a Carleman operator has been extended in many directions. By replacing $ L _ {2} $ by an arbitrary Banach function space one obtains the so-called generalized Carleman operators (see [a3]) and by considering Bochner integrals (cf. Bochner integral) and abstract Banach spaces one is lead to the so-called Carleman and Korotkov operators on a Banach space ([a4]).

References

[a1] K. Maurin, "Methods of Hilbert spaces" , PWN (1967)
[a2] P.R. Halmos, V.S. Sunder, "Bounded integral operators on -spaces" , Ergebnisse der Mathematik und ihrer Grenzgebiete , 96 , Springer (1978)
[a3] A.R. Schep, "Generalized Carleman operators" Indagationes Mathematicae , 42 (1980) pp. 49–59
[a4] N. Gretsky, J.J. Uhl, "Carleman and Korotkov operators on Banach spaces" Acta Sci. Math , 43 (1981) pp. 111–119
How to Cite This Entry:
Carleman operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carleman_operator&oldid=19140
This article was adapted from an original article by A.R. Schep (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article