Carleman operator

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A Carleman operator on the space is an integral operator , i.e., a.e. for , such that a.e. on . 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 -spaces is [a2]. The notion of a Carleman operator has been extended in many directions. By replacing 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]).


[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
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Carleman operator. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.R. Schep (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article