Difference between revisions of "Carleman operator"
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+ | A Carleman operator on the space $ L _ {2} ( X, \mu ) $ | ||
+ | is an [[Integral operator|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|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> |
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 |
Carleman operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carleman_operator&oldid=19140