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Carleman operator

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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=46217
This article was adapted from an original article by A.R. Schep (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article